用户:CaffeineP/笔记/数学分析 (伍胜健)

数学分析^w

第一章 函数

1.1 实数

1.1.4 几个常用不等式

三角不等式

${\underset {x\in \mathbb {R} ,\,y\in \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}{\mathit {true}}\,\Rightarrow \color {CadetBlue}\left\{\color {black}{\begin{alignedat}{2}&\!\!{}\pm x&&{}\leqslant |x|\\&\!\!{}\pm y&&{}\leqslant |y|\end{alignedat}}\right.\color {black}\Rightarrow {\begin{aligned}\pm (x+y)\\\pm (x-y)\end{aligned}}\!\leqslant |x|\color {RoyalBlue}+\color {black}|y|\,\Rightarrow \,{\color {RoyalBlue}\left|\color {black}x\pm y\right|}\leqslant |x|+|y|\right)$

${\underset {x\in \mathbb {R} ,\,y\in \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}{\mathit {true}}\,\Rightarrow \color {CadetBlue}\left\{\color {black}{\begin{aligned}&|{\color {RoyalBlue}x}|\leqslant |{\color {RoyalBlue}x-y}|+|{\color {RoyalBlue}y}|\\&|{\color {RoyalBlue}y}|\leqslant |{\color {RoyalBlue}y-x}|+|{\color {RoyalBlue}x}|=|{\color {RoyalBlue}x-y}|+|x|\end{aligned}}\right.\color {black}\Rightarrow \,|x-y|\geqslant \!{\begin{aligned}|x|\color {RoyalBlue}-|y|\\|y|\color {RoyalBlue}-|x|\end{aligned}}\,\Rightarrow \,|x-y|\geqslant {\color {RoyalBlue}{\big |}}|x|-|y|\color {RoyalBlue}{\big |}\right)$

${\underset {x\in \mathbb {R} ,\,y\in \mathbb {R} }{\forall }}|x+y|=|x-({\color {RoyalBlue}-y})|\geqslant {\big |}|x|-\left|{\color {RoyalBlue}-y}\right|\!{\big |}={\big |}|x|-|y|{\big |}$

$\color {darkgreen}{\underset {x\in \mathbb {R} ,\,y\in \mathbb {R} }{\forall }}{\big |}|x|-|y|{\big |}\leqslant |x{{}\color {RoyalBlue}\pm {}}y|\leqslant |x|+|y|$

Bernoulli 不等式

$\color {darkgreen}{\underset {x\geqslant -1}{\forall }}\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}{\mathit {true}}\,&\Rightarrow \color {CadetBlue}{\begin{cases}\!\color {black}{\hphantom {{\underset {n\in \mathbb {N} ^{*}}{\forall }}{\big (}}}(1+x)^{1}=1+1x\\\!{\color {black}{\underset {n\in \mathbb {N} ^{*}}{\forall }}}\!\left(\color {black}(1+x)^{n}\geqslant 1+nx\,\Rightarrow \,(1+x)^{n\color {RoyalBlue}+1}\geqslant (1+nx){\color {RoyalBlue}(1+x)}=1+(n+1)x+x^{2}\geqslant 1+(n+1)x\right)\end{cases}}\\&\Rightarrow \quad \!\color {darkgreen}{\underset {n\in \mathbb {N} ^{*}}{\forall }}\ \,(1+x)^{n}\geqslant 1+nx\end{aligned}}\!\!\!\right)$

第二章 序列的极限

2.1 序列极限的定义

2.1.2 序列极限的定义

定义2.1.1：序列极限^w和收敛序列

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}{\begin{cases}\displaystyle {\color {black}{\underset {a\in \mathbb {R} }{\forall }}}\left(\color {black}x_{n}\xrightarrow {n\to \infty } a\,\Leftrightarrow \,\lim _{n\to \infty }x_{n}=a\ :\Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {\color {RoyalBlue}k\in \mathbb {R} }{\exists }}\ {\underset {n>{\color {RoyalBlue}k},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \right)\\\color {black}\langle x_{n}\rangle {\text{ converges }}:\Leftrightarrow \,{\underset {a\in \mathbb {R} }{\exists }}\,x_{n}\xrightarrow {n\to \infty } a\end{cases}}$

例2.1.5

${\mathit {true}}\,\Rightarrow {\underset {\varepsilon >0,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}\left|{n \over n+1}-1\right|<\varepsilon \,\Leftrightarrow \,{1 \over n+1}<\varepsilon \,\Leftrightarrow \,n>{1 \over \varepsilon }-1\right)\color {black}\Rightarrow \,\color {darkgreen}{n \over n+1}\xrightarrow {n\to \infty } 1$

例2.1.6

${\begin{aligned}{\mathit {true}}\,&\Rightarrow {\underset {q\in {\color {RoyalBlue}(-1,\,0)\cup (0,\,1)}}{\forall }}\color {CadetBlue}\left(\color {black}{\mathit {true}}\,\Rightarrow {\underset {\varepsilon >0,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}|q^{n}|<\varepsilon \Leftrightarrow |q|^{n}<{\color {RoyalBlue}|q|^{\log _{|q|}\!\color {black}\varepsilon }}\Leftrightarrow n{{}\color {RoyalBlue}>{}}\log _{|q|}\!\varepsilon \right)\color {black}\Rightarrow \,q^{n}\xrightarrow {n\to \infty } 0\right)\color {black}\land \,{\color {RoyalBlue}0}^{n}\xrightarrow {n\to \infty } 0\\\\&\Rightarrow {\underset {\color {Sepia}q\in (-1,\,1)}{\forall }}\color {darkgreen}\,q^{n}\xrightarrow {n\to \infty } 0\end{aligned}}$

例2.1.7

${\begin{aligned}{\mathit {true}}\,&\Rightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}\left|{n^{2}-n+2 \over 3n^{2}+2n+4}-{\frac {1}{3}}\right|=\left|{3n^{2}-3n+6-3n^{2}-2n-4 \over {9n^{2}+6n+12}}\right|={5n-2 \over 9n^{2}+6n+12}<{5n \over 9n^{2}}<{\frac {1}{n}}\right)\\&\Rightarrow {\underset {\varepsilon >0,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}\left|{n^{2}-n+2 \over 3n^{2}+2n+4}-{\frac {1}{3}}\right|<\varepsilon \,\Leftarrow \,{\color {RoyalBlue}{\frac {1}{n}}}<\varepsilon \,\Leftrightarrow \,n>{\frac {1}{\varepsilon }}\right)\color {black}\Rightarrow \,\color {darkgreen}{n^{2}-n+2 \over 3n^{2}+2n+4}\xrightarrow {n\to \infty } {\frac {1}{3}}\end{aligned}}$

例2.1.8

证法1：${\underset {\color {Sepia}a>1}{\forall }}\color {CadetBlue}\left(\color {black}{\mathit {true}}\,\Rightarrow {\underset {\varepsilon >0,\,n\in \mathbb {N} ^{*}}{\forall }}{\color {CadetBlue}(}|{\sqrt[{n}]{a}}-1|<\varepsilon \,\Leftrightarrow \,{\sqrt[{n}]{a}}-1<\varepsilon \,\Leftrightarrow \,{\sqrt[{n}]{a}}<1+\varepsilon \,\Leftrightarrow \,n>\log _{1+\varepsilon }a{\color {CadetBlue})}\,\Rightarrow \,\color {darkgreen}{\sqrt[{n}]{a}}\xrightarrow {n\to \infty } 1\right)$

证法2：

${\underset {\color {Sepia}a>1}{\forall }}\color {CadetBlue}\left(\color {black}{\begin{aligned}{\mathit {true}}\,&\Rightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}\,a=(1+{\color {RoyalBlue}{\sqrt[{n}]{a}}-1})^{n}>n({\color {RoyalBlue}{\sqrt[{n}]{a}}-1})\\&\Rightarrow {\underset {\varepsilon >0,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}|{\sqrt[{n}]{a}}-1|<\varepsilon \,\Leftrightarrow \,{\sqrt[{n}]{a}}-1<\varepsilon \,\Leftrightarrow \,{\color {RoyalBlue}n}({\sqrt[{n}]{a}}-1)<{\color {RoyalBlue}n}\varepsilon \,\Leftarrow \,{\color {RoyalBlue}a}<n\varepsilon \,\Leftrightarrow \,n>{\frac {a}{\varepsilon }}\right)\color {black}\Rightarrow \,\color {darkgreen}{\sqrt[{n}]{a}}\xrightarrow {n\to \infty } 1\end{aligned}}\right)$

发散序列的定义

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}{\begin{aligned}\langle x_{n}\rangle \,{\text{diverges}}\,:\!&\Leftrightarrow \,{\color {RoyalBlue}\neg }\,\langle x_{n}\rangle \ {\text{converges}}\,\Leftrightarrow \,{\underset {a\in \mathbb {R} }{\color {RoyalBlue}\nexists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \\&\Leftrightarrow \,{\underset {a\in \mathbb {R} }{\color {RoyalBlue}\forall }}\ {\underset {\varepsilon >0}{\color {RoyalBlue}\exists }}\ {\underset {N\in \mathbb {N} ^{*}}{\color {RoyalBlue}\forall }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\color {RoyalBlue}\exists }}|x_{n}-a|{{}\color {RoyalBlue}\geqslant {}}\varepsilon \ \Leftrightarrow \,{\underset {a\in \mathbb {R} }{\color {RoyalBlue}\forall }}\ {\underset {\varepsilon >0}{\color {RoyalBlue}\exists }}\ {\underset {k\in \mathbb {R} }{\color {RoyalBlue}\forall }}\ {\underset {n>k,\,n\in \mathbb {N} ^{*}}{\color {RoyalBlue}\exists }}|x_{n}-a|{{}\color {RoyalBlue}\geqslant {}}\varepsilon \end{aligned}}\right)$

例2.1.9：奇数项收敛，偶数项发散

$\color {Sepia}x_{n}:={\begin{cases}{\frac {1}{n}},&n=1,\,3,\,5,\cdots \\n,&n=2,\,4,\,6,\cdots \end{cases}}$

${\begin{aligned}{\mathit {true}}\,&\Rightarrow {\underset {a\in \mathbb {R} ,\,\varepsilon >0,\,N\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}{\begin{aligned}{\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \,&\Rightarrow {\underset {n>N,\,n\in \color {RoyalBlue}\{2,\,4,\,6,\,...\}}{\forall }}|{\color {RoyalBlue}n}-a|<\varepsilon \\&\Rightarrow \,(N,\,+\infty )\cap \{2,\,4,\,6,\,...\}\subseteq (a-\varepsilon ,\,a+\varepsilon )\,\Rightarrow \,{\mathit {false}}\end{aligned}}\right)\\\\&\Rightarrow {\underset {a\in \mathbb {R} }{\nexists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \,\Rightarrow \,\color {darkgreen}\langle x_{n}\rangle \,{\text{diverges}}\end{aligned}}$

2.1.3 无穷小量

定义2.1.2：无穷小量^w

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}x_{n}=o(1)(n\to \infty )\,:\Leftrightarrow \,x_{n}\xrightarrow {n\to \infty } {\color {RoyalBlue}0}\ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|<\varepsilon \ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {k\in \mathbb {R} }{\exists }}\ {\underset {n>k,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|<\varepsilon \right)$

例2.1.10：阶乘比指数发散得更快

${\underset {\color {Sepia}a>1}{\forall }}\color {CadetBlue}\left(\color {black}{\begin{aligned}{\mathit {true}}\,&\Rightarrow {\underset {{\color {RoyalBlue}n>[a]},\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}\left|{a^{n} \over n!}\right|={a^{n} \over n!}={a^{[a]} \over [a]!}\cdot {a \over [a]+1}\cdots {\frac {a}{n}}\leqslant {a^{[a]} \over [a]!}\cdot {\frac {a}{n}}\right)\\&\Rightarrow {\underset {\varepsilon >0,\,{\color {RoyalBlue}n>[a]},\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}\left|{a^{n} \over n!}\right|<\varepsilon \,\Leftarrow \,{\color {RoyalBlue}{a^{[a]} \over [a]!}\cdot {\frac {a}{n}}}<\varepsilon \,\Leftrightarrow \,n>{a^{[a]+1} \over [a]!\varepsilon }\right)\color {black}\Rightarrow \,\color {darkgreen}{a^{n} \over n!}\xrightarrow {n\to \infty } 0\end{aligned}}\right)$

例2.1.11：奇数项和偶数项以不同速率收敛

$\color {Sepia}x_{n}:={\begin{cases}{1 \over 2^{n}},&n=1,\,3,\,5,\cdots \\{\frac {1}{\sqrt {n}}},&n=2,\,4,\,6,\cdots \end{cases}}\color {black},\quad {\begin{aligned}{\underset {n\in \mathbb {N} ^{*}}{\forall }}\end{aligned}}0<x_{n}\leqslant {\frac {1}{\sqrt {n}}}$

${\mathit {true}}\Leftrightarrow {\underset {\varepsilon >0,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}|x_{n}|<\varepsilon \Leftarrow {\color {RoyalBlue}{\frac {1}{\sqrt {n}}}}<\varepsilon \Leftrightarrow n>{\frac {1}{\varepsilon ^{2}}}\right)\color {black}\Rightarrow \,\color {darkgreen}x_{n}\xrightarrow {n\to \infty } 0$

定理2.1.1：无穷小量的性质

（1）序列是无穷小量当且仅当其绝对值是无穷小量

$\color {darkgreen}{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\!\left(\color {darkgreen}x_{n}\xrightarrow {n\to \infty } 0\ \color {black}\Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|<\varepsilon \ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}{\big |}{\color {RoyalBlue}|}x_{n}{\color {RoyalBlue}|}{\big |}<\varepsilon \color {darkgreen}\,\Leftrightarrow \,|x_{n}|\xrightarrow {n\to \infty } 0\right)$

（2）无穷小量的常数倍是无穷小量

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\!\color {CadetBlue}\left(\color {black}{\begin{aligned}&\color {Sepia}\,{\overset {\phantom {f}}{x_{n}}}\xrightarrow {n\to \infty } 0\\\Rightarrow \,&\color {CadetBlue}{\begin{cases}\color {black}0x_{n}=0\xrightarrow {n\to \infty } 0\\\!{\color {black}{\begin{aligned}{\underset {M\in \mathbb {R} }{\underset {M\neq 0}{\forall }}}\end{aligned}}}\!\left(\color {black}{\begin{aligned}{\mathit {true}}\,&\Rightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|<\varepsilon \ \Rightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|<\color {RoyalBlue}{\varepsilon \over |M|}\\&\Rightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|Mx_{n}|<\varepsilon \,\Rightarrow \,Mx_{n}\xrightarrow {n\to \infty } 0\end{aligned}}\right)\end{cases}}\!\!\!\\\Rightarrow \,&\color {darkgreen}\,{\underset {M\in \mathbb {R} }{\forall }}\,Mx_{n}\xrightarrow {n\to \infty } 0\end{aligned}}\right)$

（3）极限与无穷小量的关系

$\color {darkgreen}{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\!\color {CadetBlue}\left(\color {darkgreen}x_{n}\xrightarrow {n\to \infty } a\ \color {black}\Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \,\color {darkgreen}\Leftrightarrow \,x_{n}-a\xrightarrow {n\to \infty } 0\right)$

2.1.4 无穷大量

定义2.1.3：（正、负）无穷大量

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}{\begin{cases}\color {black}\displaystyle x_{n}\xrightarrow {n\to \infty } +\infty \ :\Leftrightarrow \,{\underset {M>0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}\ x_{n}>{\phantom {-}}M\ \Leftrightarrow \,{\underset {M>0}{\forall }}\ {\underset {\color {RoyalBlue}k\in \mathbb {R} }{\exists }}\ {\underset {n>{\color {RoyalBlue}k},\,n\in \mathbb {N} ^{*}}{\forall }}\ x_{n}>{\phantom {-}}M\\\color {black}\displaystyle x_{n}\xrightarrow {n\to \infty } -\infty \ :\Leftrightarrow \,{\underset {M>0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}\ x_{n}<-M\ \Leftrightarrow \,{\underset {M>0}{\forall }}\ {\underset {\color {RoyalBlue}k\in \mathbb {R} }{\exists }}\ {\underset {n>{\color {RoyalBlue}k},\,n\in \mathbb {N} ^{*}}{\forall }}\ x_{n}<-M\\\color {black}{\begin{aligned}\!x_{n}\xrightarrow {n\to \infty } \ \ \,\infty \ :\!&\Leftrightarrow \ |x_{n}|\xrightarrow {n\to \infty } +\infty \ \Leftrightarrow \,{\underset {M>0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}\ |x_{n}|>M\ \Leftrightarrow \,{\underset {M>0}{\forall }}\ {\underset {k\in \mathbb {R} }{\exists }}\ {\underset {n>k,\,n\in \mathbb {N} ^{*}}{\forall }}\ |x_{n}|>M\\&{{}\color {RoyalBlue}\Leftarrow }\ \ x_{n}\xrightarrow {n\to \infty } +\infty \,\lor \,x_{n}\xrightarrow {n\to \infty } -\infty \end{aligned}}\end{cases}}$

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\exists }}\color {CadetBlue}\left(\color {black}{\overset {\phantom {f}}{x_{n}}}\xrightarrow {n\to \infty } \ \ \,\infty \ \ \,{{}\color {RoyalBlue}\nRightarrow }\ \ x_{n}\xrightarrow {n\to \infty } +\infty \,\lor \,x_{n}\xrightarrow {n\to \infty } -\infty \right)$

例2.1.12：调和级数^w是正无穷大量

${\begin{aligned}\color {Sepia}{\underset {n\in \mathbb {N} ^{*}}{\forall }}\,x_{n}:={}&{\color {Sepia}1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}}\geqslant 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{\color {RoyalBlue}2^{[\log _{2}n]}}}\\\geqslant {}&1+{\frac {1}{2}}\times 1+{\frac {1}{4}}\times 2+\cdots +{\frac {1}{2^{[\log _{2}n]}}}\times 2^{[\log _{2}n]-1}=1+{\frac {[\log _{2}n]}{2}}>{\frac {\log _{2}n}{2}}\end{aligned}}$

${\mathit {true}}\,\Rightarrow {\underset {M>0,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}x_{n}>M\Leftarrow {\color {RoyalBlue}{\frac {\log _{2}n}{2}}}>M\Leftrightarrow n>2^{2M}\right)\color {black}\Rightarrow \,\color {darkgreen}x_{n}\xrightarrow {n\to \infty } +\infty$

另见#例2.4.8：用 Cauchy 收敛准则证明调和级数发散。

定理2.1.2：无穷大量和无穷小量的关系

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {Sepia}{\begin{array}{c}\ \forall _{n\in \mathbb {N} ^{*}}\\x_{n}\neq 0\end{array}}\color {black}\Rightarrow \color {CadetBlue}\left(\color {black}{\begin{aligned}\color {darkgreen}x_{n}\xrightarrow {n\to \infty } 0\ &\Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|<\varepsilon \ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}\left|{\frac {1}{x_{n}}}\right|>{\frac {1}{\varepsilon }}\\&\Leftrightarrow \,{\underset {{\color {RoyalBlue}M}>0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}\left|{\frac {1}{x_{n}}}\right|>\color {RoyalBlue}M\,\color {darkgreen}\Leftrightarrow \,{\frac {1}{x_{n}}}\xrightarrow {n\to \infty } \infty \end{aligned}}\right)\!\right)$

2.2 序列极限的性质

定义2.2.1：有界^w序列

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}{\begin{cases}\!\color {black}\langle x_{n}\rangle \,{\text{is bounded }}\Leftrightarrow \ x_{n}=O(1)\ :\Leftrightarrow \,{\underset {M>0}{\exists }}\ {\underset {n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|\leqslant M\ \Leftrightarrow \ \{x_{n}|n\in \mathbb {N} ^{*}\}{\text{ is bounded}}\\\color {black}x_{n}=O_{0}(1)\ :\Leftrightarrow \,{\underset {M_{2}>M_{1}>0,\,N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}M_{1}<|x_{n}|<M_{2}\ \Leftrightarrow \,{\underset {M_{2}>M_{1}>0,\,\color {RoyalBlue}k\in \mathbb {R} }{\exists }}\ {\underset {n>{\color {RoyalBlue}k},\,n\in \mathbb {N} ^{*}}{\forall }}M_{1}<|x_{n}|<M_{2}\end{cases}}$

定理2.2.1

（1）改变序列的有限多项不改变其敛散性；

改变收敛序列有限多项不改变其极限（可推出：无穷小量改变有限多项后还是无穷小量）；

（同理可证）（正、负）无穷大量改变有限多项后还是（正、负）无穷大量

${\begin{aligned}{\underset {N_{0}\in \mathbb {N} ^{*}}{\underset {\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}}\end{aligned}}\!\color {CadetBlue}\left({\color {Sepia}{\begin{array}{c}\forall \ _{n\in \mathbb {N} ^{*}}^{n>N_{0}}\\x_{n}=y_{n}\end{array}}\!\color {black}\Rightarrow }\,{\begin{cases}\!\left(\color {black}{\begin{aligned}&\color {darkgreen}\langle x_{n}\rangle {\text{ converges}}\\\Leftrightarrow \,\,&{\underset {a\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \ \Leftrightarrow \,{\underset {a\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>{\color {RoyalBlue}\max\{N_{0},{\color {black}N}\}},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \\\Leftrightarrow \,\,&{\underset {a\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>\max\{N_{0},N\},\,n\in \mathbb {N} ^{*}}{\forall }}|{\color {RoyalBlue}y_{n}}-a|<\varepsilon \ \Leftrightarrow \,{\underset {a\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>{\color {RoyalBlue}N},\,n\in \mathbb {N} ^{*}}{\forall }}|y_{n}-a|<\varepsilon \\\color {darkgreen}\Leftrightarrow \,\,&\color {darkgreen}\langle y_{n}\rangle {\text{ converges}}\end{aligned}}\right)\\{\color {darkgreen}{\underset {a\in \mathbb {R} }{\forall }}}\left(\color {black}{\begin{aligned}&\color {darkgreen}{\overset {\phantom {f}}{x_{n}}}\xrightarrow {n\to \infty } a\\\Leftrightarrow \,\,&{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {R} }{\exists }}\ {\underset {n>{\color {RoyalBlue}\max\{N_{0},{\color {black}N}\}},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \\\Leftrightarrow \,\,&{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {R} }{\exists }}\ {\underset {n>\max\{N_{0},N\},\,n\in \mathbb {N} ^{*}}{\forall }}|{\color {RoyalBlue}y_{n}}-a|<\varepsilon \ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>{\color {RoyalBlue}N},\,n\in \mathbb {N} ^{*}}{\forall }}|y_{n}-a|<\varepsilon \\\color {darkgreen}\Leftrightarrow \,\,&\color {darkgreen}y_{n}\xrightarrow {n\to \infty } a\end{aligned}}\right)\\\color {darkgreen}{\overset {\phantom {f}}{x_{n}}}\xrightarrow {n\to \infty } +\infty \ \Leftrightarrow \ y_{n}\xrightarrow {n\to \infty } +\infty \\\color {darkgreen}{\overset {\phantom {f}}{x_{n}}}\xrightarrow {n\to \infty } -\infty \ \Leftrightarrow \ y_{n}\xrightarrow {n\to \infty } -\infty \\\color {darkgreen}{\overset {\phantom {f}}{x_{n}}}\xrightarrow {n\to \infty } {\phantom {+}}\infty \ \Leftrightarrow \ y_{n}\xrightarrow {n\to \infty } {\phantom {+}}\infty \end{cases}}\!\!\right)$

（2）收敛序列极限的唯一性

${\begin{aligned}{\underset {\color {Sepia}a<b}{\underset {a\in \mathbb {R} ,\,b\in \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}}\end{aligned}}\!\color {CadetBlue}\left(\color {black}{\begin{aligned}\color {Sepia}x_{n}\xrightarrow {n\to \infty } a\ \land \ x_{n}\xrightarrow {n\to \infty } b\ &\Rightarrow {\underset {N_{1}\in \mathbb {N} ^{*},\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\!\color {CadetBlue}\left(\color {black}{\underset {n>N_{1},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<{\color {RoyalBlue}{\frac {b-a}{2}}}\ \land \,{\underset {n>N_{2},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-b|<\color {RoyalBlue}{\frac {b-a}{2}}\right)\\&\Rightarrow {\underset {N_{1}\in \mathbb {N} ^{*},\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>{\color {RoyalBlue}\max }\{N_{1},N_{2}\},\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}<{\frac {a+b}{2}}<x_{n}\,\Rightarrow \,\color {darkgreen}{\mathit {false}}\end{aligned}}\!\!\right)$

（3）收敛序列的有界性

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} ,\,a\in \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {Sepia}x_{n}\xrightarrow {n\to \infty } a\color {black}\ \Rightarrow {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<{\color {RoyalBlue}1}\ \Rightarrow \color {darkgreen}{\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|\leqslant \max\{|x_{1}|,\cdots ,|x_{N}|,|a|+1\}\right)$

定理2.2.2：收敛序列的保序性

（1）

${\begin{aligned}{\underset {\color {Sepia}a<b,\,c\in (a,\,b)}{\underset {a\in \mathbb {R} ,\,b\in \mathbb {R} }{\underset {\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}}}\end{aligned}}\color {CadetBlue}\!\!\left(\color {black}{\begin{aligned}\color {Sepia}x_{n}\xrightarrow {n\to \infty } a\ \land \ y_{n}\xrightarrow {n\to \infty } b\ &\Rightarrow {\underset {N_{1}\in \mathbb {N} ^{*}\!,\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\!\color {CadetBlue}\left(\color {black}{\underset {n>N_{1},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<{\color {RoyalBlue}c-a}\ \land \,{\underset {n>N_{2},\,n\in \mathbb {N} ^{*}}{\forall }}|y_{n}-b|<\color {RoyalBlue}b-c\right)\\&\Rightarrow {\underset {N_{1}\in \mathbb {N} ^{*}\!,\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>{\color {RoyalBlue}\max }\{N_{1},N_{2}\},\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}<c<y_{n}\,\Rightarrow \color {darkgreen}{\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}<c<y_{n}\end{aligned}}\!\right)$

（2）${\begin{aligned}{\underset {a\in \mathbb {R} ,\,b\in \mathbb {R} }{\underset {\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}}\end{aligned}}\!\color {CadetBlue}\left(\color {black}{\begin{aligned}&\,\color {Sepia}{\overset {\phantom {f}}{x_{n}}}\xrightarrow {n\to \infty } a\ \land \ y_{n}\xrightarrow {n\to \infty } b\ \land \,{\underset {N\in \mathbb {N} ^{*}}{\forall }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\exists }}x_{n}\leqslant y_{n}\\\Rightarrow \,&\color {CadetBlue}\left(\color {black}a>b\ \Rightarrow {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}y_{n}<{\color {RoyalBlue}{\frac {a+b}{2}}}<x_{n}\,\Rightarrow \,{\mathit {false}}\right)\color {black}\Rightarrow \,\color {darkgreen}a\leqslant b\end{aligned}}\right)$

推论

${\begin{aligned}{\underset {a\in \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}\end{aligned}}\!\color {CadetBlue}\left(\color {black}{\begin{aligned}\color {Sepia}x_{n}\xrightarrow {n\to \infty } a\,&\Rightarrow \color {CadetBlue}\left(\color {Sepia}a>0\ \color {Black}\Rightarrow \color {darkgreen}{\underset {a'\in (0,\,a)}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}>a'\right){\color {black}{}\land {}}\left(\color {Sepia}a<0\ \color {black}\Rightarrow \color {darkgreen}{\underset {a'\in (a,\,0)}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}<a'\right)\\&\Rightarrow \color {CadetBlue}\left(\color {Sepia}a\neq 0\ \color {black}\Rightarrow \color {darkgreen}{\underset {a'\in (0,\,|a|)}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|>a'\right)\end{aligned}}\!\!\right)$

定理2.2.3：序列极限的四则运算

（1）序列极限的加减法

${\begin{aligned}{\underset {a\in \mathbb {R} ,\,b\in \mathbb {R} }{\underset {\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}}\end{aligned}}\!\color {CadetBlue}\left(\color {black}{\begin{aligned}&\color {CadetBlue}{\begin{cases}\color {Sepia}x_{n}\xrightarrow {n\to \infty } a\\\color {Sepia}{\overset {\phantom {f}}{y_{n}}}\xrightarrow {n\to \infty } b\end{cases}}\,{\color {black}{}\Rightarrow {\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {N_{1}\in \mathbb {N} ^{*},\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\end{array}}}{\begin{cases}\color {black}{\underset {n>N_{1},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<{\color {RoyalBlue}{\frac {\varepsilon }{2}}}\\\color {black}{\underset {n>N_{2},\,n\in \mathbb {N} ^{*}}{\forall }}|y_{n}-b|<\color {RoyalBlue}{\frac {\varepsilon }{2}}\end{cases}}\\\Rightarrow {}&{\underset {\varepsilon >0}{\forall }}\ {\underset {N_{1}\in \mathbb {N} ^{*},\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>{\color {RoyalBlue}\max }\{N_{1},N_{2}\},\,n\in \mathbb {N} ^{*}}{\forall }}\\&|(x_{n}\pm y_{n})-(a\pm b)|=|(x_{n}-a)\pm (y_{n}-b)|\leqslant |x_{n}-a|+|y_{n}-b|<{\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2}}=\varepsilon \\\Rightarrow {}&\,\color {darkgreen}x_{n}\pm y_{n}\xrightarrow {n\to \infty } a\pm b\end{aligned}}\right)$

（2）序列极限的乘法

${\begin{aligned}{\underset {a\in \mathbb {R} ,\,b\in \mathbb {R} }{\underset {\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}}\end{aligned}}\!\color {CadetBlue}\left(\color {black}{\begin{aligned}&{\color {CadetBlue}{\begin{cases}\color {Sepia}x_{n}\xrightarrow {n\to \infty } a\\\color {Sepia}{\overset {\phantom {f}}{y_{n}}}\xrightarrow {n\to \infty } b\end{cases}}}\,\Rightarrow {\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {N_{2}\in \mathbb {N} ^{*}}{\underset {N_{1}\in \mathbb {N} ^{*}}{\underset {N\in \mathbb {N} ^{*}}{\exists }}}}\end{array}}\color {CadetBlue}{\begin{cases}\color {black}{\underset {n>N,\ n\in \mathbb {N} ^{*}}{\forall }}\,|y_{n}|<|a|+1\\\color {black}{\underset {n>N_{1},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<{\color {RoyalBlue}{\frac {\varepsilon }{2(|a|+1)}}}\\\color {black}{\underset {n>N_{2},\,n\in \mathbb {N} ^{*}}{\forall }}|y_{n}-b|\,<\color {RoyalBlue}{\frac {\varepsilon }{2(|a|+1)}}\end{cases}}\\\Rightarrow {}&{\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*},\,N_{1}\in \mathbb {N} ^{*},\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\\{\underset {n>{\color {RoyalBlue}\max }\{N,\,N_{1},\,N_{2}\}}{\forall }}\end{array}}\!\!\color {CadetBlue}\left(\color {black}{\begin{aligned}|x_{n}y_{n}-ab|&=|x_{n}y_{n}{{}\color {RoyalBlue}-ay_{n}+ay_{n}}-ab|\leqslant |y_{n}||x_{n}-a|+|a||y_{n}-b|\\&<(|a|+1){\frac {\varepsilon }{2(|a|+1)}}+|a|{\frac {\varepsilon }{2(|a|+1)}}<\varepsilon \end{aligned}}\right)\\\Rightarrow {}&\,\color {darkgreen}{\overset {\phantom {f}}{x_{n}y_{n}}}\xrightarrow {n\to \infty } ab\end{aligned}}\!\!\!\right)$

（3）序列极限的除法

${\begin{aligned}{\mathit {true}}&\Leftrightarrow \!{\begin{aligned}{\underset {b\in \mathbb {R} }{\underset {\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}\end{aligned}}\!\!\color {CadetBlue}\left(\color {black}{\begin{aligned}\!\!{\color {CadetBlue}{\begin{cases}\color {Sepia}\displaystyle \lim _{n\to \infty }y_{n}=b\neq 0\\\color {Sepia}{\underset {n\in \mathbb {N} ^{*}}{\forall }}\ y_{n}\neq 0\end{cases}}}\!&\Rightarrow \!{\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {N_{2}\in \mathbb {N} ^{*}}{\underset {N_{1}\in \mathbb {N} ^{*}}{\exists }}}\end{array}}\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {n>N_{1},\,n\in \mathbb {N} ^{*}}{\forall }}|y_{n}|>{\color {RoyalBlue}{\frac {|b|}{2}}}\,\land \,{\underset {n>N_{2},\,n\in \mathbb {N} ^{*}}{\forall }}|y_{n}-b|<{\color {RoyalBlue}{\frac {|b|^{2}\varepsilon }{2}}}\\&\Rightarrow {\begin{aligned}{\underset {n\in \mathbb {N} ^{*}}{\underset {n>{\color {RoyalBlue}\max }\{N_{1},N_{2}\}}{\forall }}}\end{aligned}}\left|{\frac {1}{y_{n}}}-{\frac {1}{b}}\right|=\left|{\frac {y_{n}-b}{y_{n}b}}\right|={\frac {|y_{n}-b|}{|y_{n}||b|}}<{\frac {\frac {|b|^{2}\varepsilon }{2}}{{\frac {|b|}{2}}|b|}}=\varepsilon \end{aligned}}\!\right)\\&\Rightarrow \color {darkgreen}\lim _{n\to \infty }{\frac {1}{y_{n}}}={\frac {1}{b}}\end{aligned}}\!\!\right)\\&\Rightarrow \!{\begin{aligned}{\underset {a\in \mathbb {R} ,\,b\in \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} ,\,\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}\end{aligned}}\!\color {CadetBlue}\left(\color {Sepia}\lim _{n\to \infty }x_{n}=a\ \land \,\lim _{n\to \infty }y_{n}=b\neq 0\ \land \,{\underset {n\in \mathbb {N} ^{*}}{\forall }}y_{n}\neq 0\color {black}\ \Rightarrow \,{\color {darkgreen}\lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}}=\lim _{n\to \infty }x_{n}\cdot \lim _{n\to \infty }{\frac {1}{y_{n}}}\color {darkgreen}={\frac {a}{b}}\right)\end{aligned}}$

例2.2.1

（1）${\begin{aligned}{\mathit {true}}\ &\Leftrightarrow \,\lim _{n\to \infty }{\frac {1}{n}}=\lim _{n\to \infty }{\frac {1}{n^{2}}}=0\ \Rightarrow \,\lim _{n\to \infty }\left(3+{\frac {4}{n}}-{\frac {100}{n^{2}}}\right)=3\ \land \,\lim _{n\to \infty }\left(4+{\frac {5}{n}}+{\frac {10^{5}}{n^{2}}}\right)=4\\&\Rightarrow \,{\color {darkgreen}\lim _{n\to \infty }{\frac {3n^{2}+4n-100}{4n^{2}+5n+10^{5}}}}=\lim _{n\to \infty }{\frac {3+{\frac {4}{n}}-{\frac {100}{n^{2}}}}{4+{\frac {5}{n}}+{\frac {10^{5}}{n^{2}}}}}={\frac {{\underset {n\to \infty }{\lim }}\left(3+{\frac {4}{n}}-{\frac {100}{n^{2}}}\right)}{{\underset {n\to \infty }{\lim }}\left(4+{\frac {5}{n}}+{\frac {10^{5}}{n^{2}}}\right)}}\color {darkgreen}={\frac {3}{4}}\end{aligned}}$

（2）无穷等比级数

${\underset {\color {Sepia}q\in (-1,\,1)}{\forall }}\color {CadetBlue}\left(\color {black}{\mathit {true}}\Leftrightarrow \lim _{n\to \infty }q^{n+1}=0\Rightarrow {\color {darkgreen}\lim _{n\to \infty }(1+q+\cdots +q^{n})}=\lim _{n\to \infty }\color {RoyalBlue}{\frac {1-q^{n+1}}{1-q}}\color {darkgreen}={\frac {1}{1-q}}\right)$

定理2.2.4：序列的夹逼收敛原理

$\!{\begin{aligned}{\underset {\ a\in \mathbb {R} ,\,N_{0}\in \mathbb {N} ^{*}}{\underset {\langle z_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\underset {\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}}}\end{aligned}}\!\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}&\color {Sepia}\lim _{n\to \infty }x_{n}=\lim _{n\to \infty }y_{n}=a\ \land \,{\underset {n>N_{0},\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}\leqslant z_{n}\leqslant y_{n}\\\Rightarrow {}&{\underset {\varepsilon >0}{\forall }}\ {\underset {N_{1}\in \mathbb {N} ^{*},\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\color {CadetBlue}\left(\color {black}{\underset {n>N_{0},\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}\leqslant z_{n}\leqslant y_{n}\,\land \,{\underset {n>N_{1},\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\varepsilon \,\land \,{\underset {n>N_{2},\,n\in \mathbb {N} ^{*}}{\forall }}|y_{n}-a|<\varepsilon \right)\\\Rightarrow {}&{\underset {\varepsilon >0}{\forall }}\ {\underset {N_{1}\in \mathbb {N} ^{*},\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>{\color {RoyalBlue}\max }\{N_{0},N_{1},N_{2}\},\,n\in \mathbb {N} ^{*}}{\forall }}{\color {CadetBlue}(}{\mathit {true}}\Leftrightarrow a-\varepsilon <x_{n}\leqslant z_{n}\leqslant y_{n}<a+\varepsilon \Rightarrow |z_{n}-a|<\varepsilon {\color {CadetBlue})}\\\Rightarrow {}&\color {darkgreen}\lim _{n\to \infty }z_{n}=a\end{aligned}}\!\!\right)$

例2.2.2

（1）

（2）

例2.2.5

（1）

（2）

（3）

2.3 单调收敛原理

2.3.1 单调收敛原理

定理2.3.1（单调收敛原理）：单调有界序列必收敛

使用：确界存在定理

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}{\begin{aligned}&\color {Sepia}{\underset {M\in \mathbb {R} }{\exists }}x_{1}\leqslant x_{2}\leqslant \cdots \leqslant M\\\Rightarrow &\color {CadetBlue}\left(\color {black}{\begin{aligned}{\mathit {true}}&\Leftrightarrow \{x_{n}|n\in \mathbb {N} ^{*}\}{\text{ is nonempty and bounded above}}\Leftrightarrow \{x_{n}|n\in \mathbb {N} ^{*}\}{\text{ has a lub}}\\&\Leftrightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}x_{n}\leqslant \sup _{k\in \mathbb {N} ^{*}}x_{k}\land {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}x_{N}>\sup _{k\in \mathbb {N} ^{*}}x_{k}-\varepsilon \\&\Rightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}{\mathit {true}}\Leftrightarrow \sup _{k\in \mathbb {N} ^{*}}x_{k}-\varepsilon <x_{N}\leqslant x_{n}\leqslant \sup _{k\in \mathbb {N} ^{*}}x_{k}\Rightarrow \left|x_{n}-\sup _{k\in \mathbb {N} ^{*}}x_{k}\right|<\varepsilon \right)\\&\Rightarrow \color {darkgreen}\lim _{n\to \infty }x_{n}=\sup _{n\in \mathbb {N} ^{*}}x_{n}\end{aligned}}\right)\end{aligned}}\right)$

同理，${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {Sepia}{\underset {M\in \mathbb {R} }{\exists }}x_{1}\geqslant x_{2}\geqslant \cdots \geqslant M\color {black}\Rightarrow \color {darkgreen}\lim _{n\to \infty }x_{n}=\inf _{n\in \mathbb {N} ^{*}}x_{n}\right)$

单调序列必广义收敛

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}{\begin{cases}\color {Sepia}x_{1}\leqslant x_{2}\leqslant \cdots \land {\overset {}{\underset {M>0}{\forall }}}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}x_{N}>{\phantom {-}}M\color {black}\Rightarrow {\underset {M>0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}{\phantom {-}}M<x_{N}\leqslant x_{N+1}\leqslant \cdots \Rightarrow \color {darkgreen}\displaystyle \lim _{n\to \infty }x_{n}=+\infty \\\color {Sepia}x_{1}\geqslant x_{2}\geqslant \cdots \land {\underset {M>0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}x_{N}<-M\color {black}\Rightarrow {\underset {M>0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}{-M}>x_{N}\geqslant x_{N+1}\geqslant \cdots \Rightarrow \color {darkgreen}\displaystyle \lim _{n\to \infty }x_{n}=-\infty \end{cases}}$

2.3.2 无理数 e ^w 和欧拉常数 c

使用：Bernoulli 不等式，单调收敛原理

${\underset {n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}x_{n}:=\left(1+{\frac {1}{n}}\right)^{n},\,y_{n}:=\left(1+{\frac {1}{n}}\right)^{n+1}>x_{n}\right)$

${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}{\begin{cases}\!\color {black}{\begin{alignedat}{4}{\frac {x_{n+1}}{x_{n}}}&=\left({\frac {n+2}{n+1}}\right)^{n+1}\left({\frac {n}{n+1}}\right)^{n}&&=\left({\frac {(n+2)\color {RoyalBlue}n}{(n+1)\color {RoyalBlue}^{2}}}\right)^{n+1}\color {RoyalBlue}{\frac {n+1}{n}}&&=\left(1-{\frac {1}{(n+1)^{2}}}\right)^{n+1}{\frac {n+1}{n}}\\&\geqslant \left(1-{\frac {1}{n+1}}\right){\frac {n+1}{n}}&&=1\\{\frac {y_{n}}{y_{n+1}}}&=\left({\frac {n+1}{n}}\right)^{n+1}\left({\frac {n+1}{n+2}}\right)^{n+2}&&=\left({\frac {(n+1)\color {RoyalBlue}^{2}}{{\color {RoyalBlue}n}(n+2)}}\right)^{n+2}\color {RoyalBlue}{\frac {n}{n+1}}&&=\left(1+{\frac {1}{n(n+2)}}\right)^{n+2}{\frac {n}{n+1}}\\&\geqslant \left(1+{\frac {1}{n}}\right){\frac {n}{n+1}}&&=1\end{alignedat}}\end{cases}}\\&\Rightarrow x_{1}\leqslant x_{2}\leqslant \cdots <\cdots \leqslant y_{2}\leqslant y_{1}\Rightarrow \lim _{n\to \infty }x_{n}=\sup _{n\in \mathbb {N} ^{*}}x_{n}\land \lim _{n\to \infty }y_{n}=\inf _{n\in \mathbb {N} ^{*}}y_{n}\end{aligned}}$

${\mathit {true}}\Leftrightarrow {\frac {\displaystyle \lim _{n\to \infty }y_{n}}{\displaystyle \lim _{n\to \infty }x_{n}}}={\lim _{n\to \infty }{\frac {y_{n}}{x_{n}}}}={\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)}=1\Rightarrow \color {darkgreen}{\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}={\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n+1}}=:e$

${\underset {x>0}{\forall }}\ln x:=\log _{e}x$

2.4 实数系连续性的基本定理

2.4.1 闭区间套定理

定理2.4.1（闭区间套定理）

使用：单调收敛原理

${\begin{aligned}{\underset {\langle b_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\underset {\langle a_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}\left(\color {black}{\begin{aligned}&\color {Sepia}{\underset {n\in \mathbb {N} ^{*}}{\forall }}{\color {CadetBlue}(}a_{n}\leqslant b_{n}\land [a_{n},\,b_{n}]\supseteq [a_{n+1},\,b_{n+1}]{\color {CadetBlue})}\land \lim _{n\to \infty }(b_{n}-a_{n})=0\\\Leftrightarrow {}&\ a_{1}\leqslant a_{2}\leqslant \cdots \leqslant \cdots \leqslant b_{2}\leqslant b_{1}\land \lim _{n\to \infty }(b_{n}-a_{n})=0\\\Rightarrow {}&\lim _{n\to \infty }a_{n}=\sup _{n\in \mathbb {N} ^{*}}a_{n}\land \lim _{n\to \infty }b_{n}=\inf _{n\in \mathbb {N} ^{*}}b_{n}\land \lim _{n\to \infty }b_{n}-\lim _{n\to \infty }a_{n}=0\Rightarrow \sup _{n\in \mathbb {N} ^{*}}a_{n}=\inf _{n\in \mathbb {N} ^{*}}b_{n}\\\Rightarrow {}&{\color {darkgreen}{\underset {n\in \mathbb {N} ^{*}}{\bigcup }}[a_{n},\,b_{n}]}=\left\{c\in \mathbb {R} {\Big |}{\underset {n\in \mathbb {N} ^{*}}{\forall }}a_{n}\leqslant c\leqslant b_{n}\right\}=\left[\sup _{n\in \mathbb {N} ^{*}}a_{n},\,+\infty \right)\cap \left(-\infty ,\,\inf _{n\in \mathbb {N} ^{*}}b_{n}\right]\\&{\hphantom {{\underset {n\in \mathbb {N} ^{*}}{\bigcup }}[a_{n},\,b_{n}]}}\color {darkgreen}=\left\{\sup _{n\in \mathbb {N} ^{*}}a_{n}\right\}=\left\{\inf _{n\in \mathbb {N} ^{*}}b_{n}\right\}\end{aligned}}\right)$

2.4.2 有限覆盖定理

定理2.4.2（有限覆盖定理）：闭区间的开覆盖必包含有限覆盖

使用：闭区间套定理

${\begin{aligned}{\underset {\{E_{\lambda }\}_{\lambda \in \Lambda }}{\underset {\Lambda }{\underset {\color {Sepia}a\leqslant b}{\underset {b\in \mathbb {R} }{\underset {a\in \mathbb {R} }{\forall }}}}}}\end{aligned}}\!\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}&\color {Sepia}{\underset {\lambda \in \Lambda }{\forall }}E_{\lambda }{\text{ is an open interval }}\land \,[a,\,b]\subseteq {\underset {\lambda \in \Lambda }{\bigcup }}E_{\lambda }\\\Rightarrow &\color {CadetBlue}\left(\!\color {black}{\begin{aligned}&{\underset {N\in \mathbb {N} ^{*},\,\{\lambda _{1},\cdots ,\lambda _{N}\}\subseteq \Lambda }{\nexists }}[a,\,b]\subseteq \bigcup _{j=1}^{N}E_{\lambda _{j}}\\\Rightarrow &\color {CadetBlue}\left(\!\color {black}{\begin{aligned}&{\begin{alignedat}{3}&[a_{0},\,b_{0}]&&:=\quad \,[a,\,b],\\{\underset {n\in \mathbb {N} }{\forall }}&[a_{n+1},\,b_{n+1}]&&:={\begin{cases}\left[a_{n},\,\color {RoyalBlue}{\frac {a_{n}+b_{n}}{2}}\right],&{\underset {N\in \mathbb {N} ^{*},\,\{\lambda _{1},\cdots ,\lambda _{N}\}\subseteq \Lambda }{\nexists }}\left[a_{n},\,{\frac {a_{n}+b_{n}}{2}}\right]\subseteq \bigcup _{j=1}^{N}E_{\lambda _{j}}\\\left[{\color {RoyalBlue}{\frac {a_{n}+b_{n}}{2}}},\,b_{n}\right],&{\text{else}}\end{cases}},\end{alignedat}}\\&{\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {n\in \mathbb {N} }{\forall }}\color {CadetBlue}{\begin{cases}\color {black}[a_{n},\,b_{n}]\supseteq [a_{n+1},\,b_{n+1}]\\\color {black}b_{n}-a_{n}={\frac {b-a}{2^{n}}}\rightarrow 0\,(n\to \infty )\\\color {black}{\underset {N\in \mathbb {N} ^{*},\,\{\lambda _{1},\cdots ,\lambda _{N}\}\subseteq \Lambda }{\nexists }}[a_{n},\,b_{n}]\subseteq \bigcup _{j=1}^{N}E_{\lambda _{j}}\end{cases}}\\&\Rightarrow \color {CadetBlue}\left(\!\color {black}{\begin{alignedat}{3}{\mathit {true}}&\Leftrightarrow \color {CadetBlue}\left\{\color {black}{\begin{aligned}&\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=:\xi \\&\{\xi \}={\underset {n\in \mathbb {N} }{\bigcap }}[a_{n},\,b_{n}]\\&{\underset {c\in \mathbb {R} ,\,d\in \mathbb {R} ,\,\lambda _{0}\in \Lambda }{\exists }}\xi \in E_{\lambda _{0}}=(c,\,d)\end{aligned}}\right\}&&{\color {black}{}\Rightarrow {\begin{aligned}{\underset {n\in \mathbb {N} }{\underset {\lambda _{0}\in \Lambda }{\underset {c\in \mathbb {R} ,\,d\in \mathbb {R} }{\exists }}}}\end{aligned}}}{\begin{cases}\color {black}E_{\lambda _{0}}=(c,\,d)\\\color {black}c<a_{n}\leqslant \xi \leqslant b_{n}<d\end{cases}}\\&\Rightarrow {\underset {\lambda _{0}\in \Lambda ,\,n\in \mathbb {N} }{\exists }}[a_{n},\,b_{n}]\subset E_{\lambda _{0}}&&\!\Leftrightarrow {\mathit {false}}\end{alignedat}}\!\right)\end{aligned}}\end{aligned}}\!\!\right)\end{aligned}}\!\right)\\\Rightarrow &\,\color {darkgreen}{\underset {N\in \mathbb {N} ^{*},\,\{\lambda _{1},\cdots ,\lambda _{N}\}\subseteq \Lambda }{\exists }}[a,\,b]\subseteq \bigcup _{j=1}^{N}E_{\lambda _{j}}\end{aligned}}\!\right)$

例2.4.3：[0, 1] 中的实数不能排成序列，即 [0, 1] 不是可数集

使用：无穷等比级数，有限覆盖定理

${\underset {l\in \mathbb {N} ^{*},\,\{n_{1},\cdots ,n_{l}\}\subset \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}{\color {RoyalBlue}{\underset {\{i,j\}\subseteq \{1,\cdots ,l\},\,i\neq j}{\forall }}n_{i}\neq n_{j}}\Rightarrow \sum _{k=1}^{l}{\frac {1}{2^{n_{k}}}}<\sum _{k=1}^{\color {RoyalBlue}\infty }{\frac {1}{2\color {RoyalBlue}^{k}}}={\frac {1}{2}}{\frac {1}{1-{\frac {1}{2}}}}=1\right)$

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}{\begin{aligned}\color {Sepia}[0,\,1]=\{x_{n}|n\in \mathbb {N} ^{*}\}&\Rightarrow [0,\,1]\subseteq \bigcup _{n\in \mathbb {N} ^{*}}U\left(x_{n},\color {RoyalBlue}{\frac {1}{2^{n+1}}}\right)\Rightarrow {\begin{aligned}{\underset {\{n_{1},\cdots ,n_{l}\}\subset \mathbb {N} ^{*}}{\underset {l\in \mathbb {N} ^{*}}{\exists }}}\end{aligned}}\color {CadetBlue}{\begin{cases}\color {black}{\underset {\{i,j\}\subseteq \{1,\cdots ,l\},\,i\neq j}{\forall }}n_{i}\neq n_{j}\\\color {black}[0,\,1]\subseteq \bigcup _{\color {RoyalBlue}k=1}^{\color {RoyalBlue}l}U\left(x_{\color {RoyalBlue}n_{k}},{\frac {1}{2^{{\color {RoyalBlue}n_{k}}+1}}}\right)\end{cases}}\\&\Rightarrow {\begin{aligned}{\underset {\{n_{1},\cdots ,n_{l}\}\subset \mathbb {N} ^{*}}{\underset {l\in \mathbb {N} ^{*}}{\exists }}}\end{aligned}}\color {CadetBlue}{\begin{cases}\color {black}{\underset {\{i,j\}\subseteq \{1,\cdots ,l\},\,i\neq j}{\forall }}n_{i}\neq n_{j}\\\color {black}\sum _{k=1}^{l}{\frac {1}{2^{\color {RoyalBlue}n_{k}}}}>1\end{cases}}\Leftrightarrow \color {darkgreen}{\mathit {false}}\end{aligned}}\right)$

更简单的证法（不用有限覆盖定理，但涉及可数无穷个开区间长度相加，故不确定是否正确）：

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {Sepia}[0,\,1]=\{x_{n}|n\in \mathbb {N} ^{*}\}\color {black}\Rightarrow [0,\,1]\subseteq \bigcup _{n\in \mathbb {N} ^{*}}U\left(x_{n},\color {RoyalBlue}{\frac {1}{2^{n+1}}}\right)\Rightarrow \sum _{n\in \mathbb {N} ^{*}}{\frac {1}{2^{\color {RoyalBlue}n}}}>1\Leftrightarrow \color {darkgreen}{\mathit {false}}\right)$

2.4.3 聚点原理

定义2.4.1：聚点、孤立点

${\begin{aligned}{\underset {x_{0}\in \mathbb {R} }{\underset {E\subseteq \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}{\begin{cases}\!\color {black}{\begin{aligned}x_{0}{\text{ is a cluster point of }}E&:\Leftrightarrow {\underset {\delta >0}{\forall }}\,U_{0}(x_{0},\delta )\cap E\neq \varnothing \Leftrightarrow {\underset {\delta >0}{\forall }}\,U(x_{0},\delta )\cap E{\text{ is infinite}}\\&{\hphantom {:}}\Leftrightarrow {\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to E}{\exists }}\color {CadetBlue}\left(\color {black}{\underset {i\in \mathbb {N} ^{*},\,j\in \mathbb {N} ^{*},\,i\neq j}{\forall }}x_{i}\neq x_{j}\,\land \,\lim _{n\to \infty }x_{n}=x_{0}\right)\end{aligned}}\\\color {black}x_{0}{\text{ is an isolated point of }}E:\Leftrightarrow x_{0}\in E\,\land \,x_{0}{\text{ is not a cluster point of }}E\Leftrightarrow {\overset {}{\underset {\delta >0}{\exists }}}U(x_{0},\delta )\cap E=\{x_{0}\}\end{cases}}$

例2.4.4

${\color {Sepia}E:=\left\{{\frac {1}{n}}{\bigg |}n\in \mathbb {N} ^{*}\right\}},\quad \mathbb {R} =(-\infty ,\,0]\cup (1,\,+\infty )\cup E\cup \bigcup _{n\in \mathbb {N} ^{*}}\left({\frac {1}{\color {RoyalBlue}n+1}},\,{\frac {1}{n}}\right)$

${\mathit {true}}\Leftrightarrow \color {CadetBlue}\left\{\color {black}{\begin{alignedat}{4}&{\overset {\hphantom {x\in \left({\frac {1}{n+1}},\,{\frac {1}{n}}\right)}}{\underset {x\color {RoyalBlue}<0}{\forall }}}&&U&&(x,{\color {RoyalBlue}-x})&&\cap E=\varnothing \\&{\overset {\hphantom {x\in \left({\frac {1}{n+1}},\,{\frac {1}{n}}\right)}}{\underset {\delta >0}{\forall }}}&&U_{0}&&({\color {RoyalBlue}0},\delta )&&\cap E\neq \varnothing \\&{\overset {\hphantom {x\in \left({\frac {1}{n+1}},\,{\frac {1}{n}}\right)}}{\underset {x\color {RoyalBlue}>1}{\forall }}}&&U&&(x,{\color {RoyalBlue}x-1})&&\cap E=\varnothing \\&{\overset {\hphantom {x\in \left({\frac {1}{n+1}},\,{\frac {1}{n}}\right)}}{\underset {n\in \mathbb {N} ^{*}}{\forall }}}&&U&&\!\!\left({\frac {1}{n}},\color {RoyalBlue}{\frac {1}{n}}-{\frac {1}{n+1}}\right)&&\cap E=\left\{{\frac {1}{n}}\right\}\\&\!{\begin{aligned}{\underset {x\in \left({\frac {1}{n+1}},\,{\frac {1}{n}}\right)}{\underset {n\in \mathbb {N} ^{*}}{\forall }}}\end{aligned}}&&U&&\!\!\left(x,\color {RoyalBlue}\min \left\{x-{\frac {1}{n+1}},{\frac {1}{n}}-x\right\}\right)&&\cap E=\varnothing \end{alignedat}}\right\}{\color {black}\Rightarrow {}}{\begin{cases}\!\color {darkgreen}{\begin{alignedat}{3}&\{{\text{cluster }}&{\text{points of }}E\}&=\{0\}\\&\{{\text{isolated }}&{\text{points of }}E\}&=\ E\end{alignedat}}\end{cases}}$

例2.4.5

$\{{\text{cluster points of }}[0,\,1]\cap \mathbb {Q} \}=[0,\,1],\quad \{{\text{isolated points of }}[0,\,1]\cap \mathbb {Q} \}=\varnothing$

定理2.4.3（聚点原理）：R 的有界无穷子集至少有一个聚点

证法一：使用有限覆盖定理

${\begin{aligned}{\underset {\color {Sepia}E\subseteq [a,\,b]}{\underset {\color {Sepia}a\leqslant b}{\underset {a\in \mathbb {R} ,\,b\in \mathbb {R} }{\forall }}}}\end{aligned}}\color {CadetBlue}\left(\color {black}{\begin{aligned}&{\color {Sepia}E{\text{ has no cluster points}}}\Rightarrow {\underset {x\in [a,\,b]}{\forall }}\ {\underset {\delta >0}{\exists }}{\color {CadetBlue}(}U(x,\delta )\cap E\subseteq \{x\},\,\delta =:\color {RoyalBlue}\delta _{x}\color {CadetBlue})\\\Rightarrow &\color {CadetBlue}\left(\color {black}{\begin{aligned}{\mathit {true}}&\Leftrightarrow [a,\,b]\subseteq \bigcup _{x\in [a,\,b]}U(x,\delta _{x})\,\Rightarrow {\underset {m\in \mathbb {N} ^{*},\,\{x_{1},\cdots ,x_{m}\}\subset [a,\,b]}{\exists }}[a,\,b]\subseteq \bigcup _{\color {RoyalBlue}j=1}^{\color {RoyalBlue}m}U({\color {RoyalBlue}x_{j}},\delta _{\color {RoyalBlue}x_{j}})\\&\Rightarrow {\underset {m\in \mathbb {N} ^{*},\,\{x_{1},\cdots ,x_{m}\}\subset [a,\,b]}{\exists }}E\subseteq \{x_{1},\cdots ,x_{m}\}\,\Rightarrow \,\color {darkgreen}E{\text{ is finite}}\end{aligned}}\right)\end{aligned}}\right)$

证法二：使用闭区间套定理

${\begin{aligned}{\underset {\color {Sepia}E\subseteq [a,\,b]}{\underset {\color {Sepia}a\leqslant b}{\underset {a\in \mathbb {R} ,\,b\in \mathbb {R} }{\forall }}}}\end{aligned}}\color {CadetBlue}\left(\color {black}{\begin{aligned}&[a_{0},\,b_{0}]:=[a,\,b],\ {\underset {n\in \mathbb {N} }{\forall }}[a_{n+1},\,b_{n+1}]:={\begin{cases}\left[a_{n},\,\color {RoyalBlue}{\frac {a_{n}+b_{n}}{2}}\right],&\left[a_{n},\,{\frac {a_{n}+b_{n}}{2}}\right]\cap E{\text{ is infinite}}\\\left[{\color {RoyalBlue}{\frac {a_{n}+b_{n}}{2}}},\,b_{n}\right],&{\text{else}}\end{cases}},\\&{\color {Sepia}{\begin{array}{c}E{\text{ is}}\\{\text{infinite}}\end{array}}}\Rightarrow \color {CadetBlue}\left\{\color {black}{\begin{aligned}&{\underset {n\in \mathbb {N} }{\forall }}[a_{n},\,b_{n}]\cap E{\text{ is infinite}}\\&\lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=:c\\&\{c\}=\bigcap _{n\in \mathbb {N} }[a_{n},\,b_{n}]\end{aligned}}\right\}{\color {black}\Rightarrow {\begin{array}{c}{\underset {\delta >0}{\forall }}\\{\underset {n\in \mathbb {N} }{\exists }}\end{array}}}{\begin{cases}\color {black}[a_{n},\,b_{n}]\subset U(c,\delta )\\\color {black}[a_{n},\,b_{n}]\cap E{\text{ is infinite}}\end{cases}}\color {black}\Rightarrow \color {darkgreen}{\begin{array}{c}c{\text{ is a cluster}}\\{\text{point of }}E\end{array}}\end{aligned}}\right)$

子序列的定义

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} ,\,\langle n_{k}\rangle }{\forall }}\color {CadetBlue}\left(\color {black}{\begin{array}{c}\langle x_{\color {RoyalBlue}n_{k}}\rangle _{{\color {RoyalBlue}k}\in \mathbb {N} ^{*}}{\text{ is a}}\\{\text{subsequence of }}\langle x_{n}\rangle \end{array}}:\Leftrightarrow \color {CadetBlue}{\begin{cases}\color {black}\langle n_{k}\rangle \in {\mathbb {N} ^{*}}^{\mathbb {N} ^{*}}\\\color {black}n_{1}<n_{2}<\cdots \end{cases}}\color {black}\Rightarrow {\begin{array}{c}\forall _{k\in \mathbb {N} ^{*}}\\n_{k}\geqslant k\end{array}}\Rightarrow \lim _{k\to \infty }n_{k}=+\infty \right)$

定理2.4.4：收敛序列的子序列亦收敛，极限与原序列相同

${\begin{aligned}{\underset {\langle n_{k}\rangle }{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}\left(\color {black}\color {Sepia}{\begin{array}{c}\langle x_{n_{k}}\rangle _{k\in \mathbb {N} ^{*}}\\{\text{is a}}\\{\text{subsequence}}\\{\text{of }}\langle x_{n}\rangle \end{array}}\color {black}\Rightarrow \color {CadetBlue}{\begin{cases}{\color {black}{\underset {a\in \mathbb {R} }{\forall }}}\left(\color {black}{\begin{aligned}\color {Sepia}\lim _{n\to \infty }x_{n}=a&\Rightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {k>N,\,k\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}{\begin{cases}\color {black}|x_{k}-a|<\varepsilon \\\color {black}n_{k}\geqslant k>N\end{cases}}\\&\Rightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {k>N,\,k\in \mathbb {N} ^{*}}{\forall }}\ \,|x_{\color {RoyalBlue}n_{k}}-a|<\varepsilon \Leftrightarrow \color {darkgreen}\lim _{k\to \infty }x_{n_{k}}=a\end{aligned}}\right)\\\displaystyle \color {Sepia}\lim _{n\to \infty }x_{n}=+\infty \color {black}\Rightarrow \color {darkgreen}\lim _{k\to \infty }x_{n_{k}}=+\infty \\\displaystyle \color {Sepia}\lim _{n\to \infty }x_{n}=-\infty \color {black}\Rightarrow \color {darkgreen}\lim _{k\to \infty }x_{n_{k}}=-\infty \\\displaystyle \color {Sepia}\lim _{n\to \infty }x_{n}=\ \ \ \infty \color {black}\Rightarrow \color {darkgreen}\lim _{k\to \infty }x_{n_{k}}=\ \ \ \infty \end{cases}}\right)$

推论：用子序列判断序列发散

一子列发散，或二子列无相同极限，则序列发散。

例2.4.6

${\mathit {true}}\Leftrightarrow \lim _{k\to \infty }\sin {\frac {{\color {RoyalBlue}4k}\pi }{2}}=0\neq 1=\lim _{k\to \infty }\sin {\frac {({\color {RoyalBlue}4k+1})\pi }{2}}\Rightarrow \color {darkgreen}\left\langle \sin {\frac {n\pi }{2}}\right\rangle _{\!n\in \mathbb {N} ^{*}}{\text{diverges}}$

例2.4.7

${\mathit {true}}\Leftrightarrow \color {CadetBlue}{\begin{cases}\color {black}\displaystyle \lim _{k\to \infty }\left(1+{\frac {(-1)^{\color {RoyalBlue}2k}}{\color {RoyalBlue}2k}}\right)^{\color {RoyalBlue}2k}=e\\\color {black}\displaystyle \lim _{k\to \infty }\left(1+{\frac {(-1)^{\color {RoyalBlue}2k+1}}{\color {RoyalBlue}2k+1}}\right)^{\color {RoyalBlue}2k+1}=\lim _{k\to \infty }\left({\frac {1}{1+{\frac {1}{2k}}}}\right)^{2k+1}={\frac {1}{e}}\neq e\end{cases}}\color {black}\Rightarrow \color {darkgreen}\left\langle \!\left(1+{\frac {(-1)^{n}}{n}}\right)^{\!\!n}\right\rangle _{\!n\in \mathbb {N} ^{*}}\!{\text{diverges}}$

定理2.4.5（Bolzano-Weierstrass 定理）：有界序列必有收敛子序列

使用：聚点原理

${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}{\begin{cases}\color {black}{\begin{array}{c}\{x_{n}|n\in \mathbb {N} ^{*}\}\\{\text{is finite}}\end{array}}\Rightarrow {\underset {x_{0}\in \{x_{n}|n\in \mathbb {N} ^{*}\}}{\exists }}{\begin{array}{c}\langle x_{n}\rangle {\text{ has infinitely many}}\\{\text{terms equaling }}x_{0}\end{array}}\Rightarrow {\begin{array}{c}\langle x_{n}\rangle {\text{ has a converging}}\\{\text{subsequence}}\end{array}}\\\color {black}{\begin{array}{c}\{x_{n}|n\in \mathbb {N} ^{*}\}\\{\text{is bounded}}\\{\text{and infinite}}\end{array}}\Rightarrow {\begin{array}{c}\{x_{n}|n\in \mathbb {N} ^{*}\}\\{\text{has a cluster}}\\{\text{point}}\end{array}}\Rightarrow {\begin{aligned}{\underset {\langle n_{k}\rangle :\mathbb {N} ^{*}\to \mathbb {N} ^{*}}{\underset {a\in \mathbb {R} }{\exists }}}\end{aligned}}{\begin{aligned}{\underset {k\in \mathbb {N} }{\forall }}\end{aligned}}{\color {CadetBlue}{\begin{cases}\color {black}x_{n_{k}}\in U\left(a,\color {RoyalBlue}{\frac {1}{k}}\right)\\\color {black}n_{k+1}>n_{k}\end{cases}}}\Rightarrow {\begin{aligned}{\underset {\langle n_{k}\rangle :\mathbb {N} ^{*}\to \mathbb {N} ^{*}}{\underset {a\in \mathbb {R} }{\exists }}}\end{aligned}}\displaystyle \lim _{k\to \infty }x_{n_{k}}=a\end{cases}}\\&\Rightarrow \!{\begin{aligned}{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\end{aligned}}\color {CadetBlue}\left(\color {Sepia}\langle x_{n}\rangle {\text{ is bounded}}\color {black}\Rightarrow \color {darkgreen}\langle x_{n}\rangle {\text{ has a converging subsequence}}\right)\end{aligned}}$

2.4.4 Cauchy 收敛准则

定义2.4.2：Cauchy 序列

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}\langle x_{n}\rangle {\text{ is Cauchy}}:\Leftrightarrow {\begin{aligned}{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\end{aligned}}\!{\begin{aligned}{\underset {m>N,\,m\in \mathbb {N} ^{*}}{\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}}\end{aligned}}|x_{n}-x_{m}|<\varepsilon \Leftrightarrow {\begin{aligned}{\underset {\varepsilon >0}{\forall }}\ {\underset {\color {RoyalBlue}k\in \mathbb {R} }{\exists }}\end{aligned}}\!{\begin{aligned}{\underset {m>{\color {RoyalBlue}k},\,m\in \mathbb {N} ^{*}}{\underset {n>{\color {RoyalBlue}k},\,n\in \mathbb {N} ^{*}}{\forall }}}\end{aligned}}|x_{n}-x_{m}|<\varepsilon \right)$

定理2.4.6（Cauchy 收敛准则）：序列收敛当且仅当它是 Cauchy 序列

使用：三角不等式，Bolzano-Weierstrass 定理

${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\!\color {CadetBlue}\left(\color {black}\!\!\!{\begin{aligned}&{\phantom {{}\Leftrightarrow {}}}\langle x_{n}\rangle \,{\text{converges}}\Leftrightarrow {\underset {a\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<\color {RoyalBlue}{\frac {\varepsilon }{2}}\\&\Rightarrow {\begin{aligned}{\underset {a\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\end{aligned}}{\begin{aligned}{\underset {m>N,\,m\in \mathbb {N} ^{*}}{\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}}\end{aligned}}|x_{n}-x_{m}|\leqslant |x_{n}-a|+|x_{m}-a|<{\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2}}=\varepsilon \Rightarrow \langle x_{n}\rangle {\text{ is Cauchy}}\\&\Rightarrow \!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-x_{\color {RoyalBlue}N+1}|<{\color {RoyalBlue}1}\Rightarrow {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n\in \mathbb {N} ^{*}}{\forall }}|x_{n}|\leqslant \max\{|x_{1}|,\cdots ,|x_{N}|,|x_{N+1}|+1\}\\&\Rightarrow \!{\begin{aligned}{\underset {a\in \mathbb {R} }{\underset {\langle n_{k}\rangle :\mathbb {N} ^{*}\to \mathbb {N} ^{*}}{\exists }}}\end{aligned}}\!\!\!\color {CadetBlue}\left(\!\!\color {black}{\begin{aligned}{\mathit {true}}&\Leftrightarrow \color {CadetBlue}{\begin{cases}\color {black}n_{1}<n_{2}<\cdots \\\color {black}\displaystyle \lim _{k\to \infty }x_{n_{k}}=a\end{cases}}{\color {black}\Rightarrow {}}{\begin{cases}\color {black}{\underset {k\in \mathbb {N} ^{*}}{\forall }}n_{k}\geqslant k\\{\color {black}\!{\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {N\in \mathbb {N} ^{*}}{\underset {K\in \mathbb {N} ^{*}}{\exists }}}\end{array}}\!}{\begin{cases}\color {black}{\underset {k>K,\,k\in \mathbb {N} ^{*}}{\forall }}|x_{n_{k}}-a|<\color {RoyalBlue}{\frac {\varepsilon }{2}}\\\color {black}{\underset {m>N,\,m\in \mathbb {N} ^{*}}{\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}}|x_{n}-x_{m}|<\color {RoyalBlue}{\frac {\varepsilon }{2}}\end{cases}}\end{cases}}\\&\Rightarrow {\begin{aligned}&{\underset {\varepsilon >0}{\forall }}\ {\underset {K\in \mathbb {N} ^{*},\,N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*},\,k>{\color {RoyalBlue}\max }\{K,N\},\,k\in \mathbb {N} ^{*}}{\forall }}\\&|x_{n}-a|\leqslant |x_{n}-x_{n_{k}}|+|x_{n_{k}}-a|<{\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2}}=\varepsilon \end{aligned}}\!\Rightarrow \lim _{n\to \infty }x_{n}=a\end{aligned}}\!\!\right)\\&\Rightarrow \langle x_{n}\rangle {\text{ converges}}\end{aligned}}\!\!\right)\end{aligned}}\!\!\right)\\&\Rightarrow \!{\begin{aligned}{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\end{aligned}}\color {CadetBlue}\left(\color {darkgreen}\langle x_{n}\rangle \,{\text{converges}}\Leftrightarrow \langle x_{n}\rangle {\text{ is Cauchy}}\right)\end{aligned}}$

例2.4.8：用 Cauchy 收敛准则证明调和级数发散

${\mathit {true}}\Leftrightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}\sum _{j=1}^{\color {RoyalBlue}2n}{\frac {1}{j}}-\sum _{j=1}^{n}{\frac {1}{j}}=\sum _{j=n+1}^{2n}{\frac {1}{j}}\geqslant \sum _{j=n+1}^{2n}{\frac {1}{\color {RoyalBlue}2n}}={\frac {n}{2n}}={\frac {1}{2}}\Rightarrow \left\langle \sum _{j=1}^{n}{\frac {1}{j}}\right\rangle _{\!\!n\in \mathbb {N} ^{*}}\!{\begin{array}{c}{\text{is not}}\\{\text{Cauchy}}\end{array}}\Leftrightarrow \color {darkgreen}\left\langle \sum _{j=1}^{n}{\frac {1}{j}}\right\rangle _{\!\!n\in \mathbb {N} ^{*}}\!\!{\text{diverges}}$

例2.4.9

${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\begin{aligned}{\underset {n>m}{\underset {n\in \mathbb {N} ^{*},\,m\in \mathbb {N} ^{*}}{\forall }}}\end{aligned}}\left|\sum _{j=1}^{n}{\frac {1}{j^{2}}}-\sum _{j=1}^{m}{\frac {1}{j^{2}}}\right|=\sum _{j=m+1}^{n}{\frac {1}{j^{2}}}<\sum _{j=m+1}^{n}{\frac {1}{({\color {RoyalBlue}j-1})j}}=\sum _{j=m+1}^{n}\left({\frac {1}{j-1}}-{\frac {1}{j}}\right)={\frac {1}{m}}-{\frac {1}{n}}<{\frac {1}{m}}\\&\Rightarrow {\begin{aligned}{\underset {m>{\color {RoyalBlue}{\frac {1}{\varepsilon }}},\,m\in \mathbb {N} ^{*}}{\underset {\varepsilon >0,\,n>{\color {RoyalBlue}{\frac {1}{\varepsilon }}},\,n\in \mathbb {N} ^{*}}{\forall }}}\end{aligned}}\left|\sum _{j=1}^{n}{\frac {1}{j^{2}}}-\sum _{j=1}^{m}{\frac {1}{j^{2}}}\right|<{\frac {1}{\color {RoyalBlue}\min\{m,n\}}}<{\frac {1}{\frac {1}{\varepsilon }}}=\varepsilon \Rightarrow \left\langle \sum _{j=1}^{n}{\frac {1}{j^{2}}}\!\right\rangle _{\!\!n\in \mathbb {N} ^{*}}\!{\begin{array}{c}{\text{is}}\\{\text{Cauchy}}\end{array}}\Leftrightarrow \color {darkgreen}\left\langle \sum _{j=1}^{n}{\frac {1}{j^{2}}}\!\right\rangle _{\!\!n\in \mathbb {N} ^{*}}\!\!{\text{converges}}\end{aligned}}$

2.5 序列的上、下极限

使用：确界存在定理，单调收敛原理

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}{\begin{cases}\!\color {black}{\begin{alignedat}{2}&\langle x_{n}\rangle {\text{ is bounded}}&&\Rightarrow \inf _{k=1}^{\infty }x_{k}\leqslant \inf _{k=2}^{\infty }x_{k}\leqslant \cdots \leqslant \cdots \leqslant \sup _{k=2}^{\infty }x_{k}\leqslant \sup _{k=1}^{\infty }x_{k}\\&&&\Rightarrow \color {CadetBlue}{\begin{cases}\color {black}\displaystyle \lim _{n\to \infty }\inf _{k=n}^{\infty }\,x_{k}=\sup _{n=1}^{\infty }\inf _{k=n}^{\infty }x_{k}=:{\underset {n\to \infty }{\underline {\lim }}}x_{n}=\liminf _{n\to \infty }x_{n}\\\color {black}\displaystyle \lim _{n\to \infty }\sup _{k=n}^{\infty }x_{k}=\inf _{n=1}^{\infty }\sup _{k=n}^{\infty }x_{k}=:{\underset {n\to \infty }{\overline {\lim }}}x_{n}=\limsup _{n\to \infty }x_{n}\end{cases}}\end{alignedat}}\\\!\color {black}{\begin{alignedat}{3}&\{x_{n}|n\in \mathbb {N} ^{*}\}{\text{ is }}{\phantom {\text{un}}}{\text{bounded below}}&&\Rightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}\inf _{k=n}^{\infty }x_{k}\in \mathbb {R} ,\quad &&{\underset {n\to \infty }{\underline {\lim }}}x_{n}:=\sup _{n=1}^{\infty }\inf _{k=n}^{\infty }x_{k}\quad {\text{(possibly }}{+\infty }{\text{)}}\\&\{x_{n}|n\in \mathbb {N} ^{*}\}{\text{ is unbounded below}}&&\Rightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}\inf _{k=n}^{\infty }x_{k}=-\infty ,\quad &&{\underset {n\to \infty }{\underline {\lim }}}x_{n}:=-\infty \\&\{x_{n}|n\in \mathbb {N} ^{*}\}{\text{ is }}{\phantom {\text{un}}}{\text{bounded above}}&&\Rightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}\sup _{k=n}^{\infty }x_{k}\in \mathbb {R} ,\quad &&{\underset {n\to \infty }{\overline {\lim }}}x_{n}:=\inf _{n=1}^{\infty }\sup _{k=n}^{\infty }x_{k}\quad {\text{(possibly }}{-\infty }{\text{)}}\\&\{x_{n}|n\in \mathbb {N} ^{*}\}{\text{ is unbounded above}}&&\Rightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}\sup _{k=n}^{\infty }x_{k}=+\infty ,\quad &&{\underset {n\to \infty }{\overline {\lim }}}x_{n}:=+\infty \\\end{alignedat}}\\\color {black}{\underset {n\to \infty }{\underline {\lim }}}x_{n}\leqslant {\underset {n\to \infty }{\overline {\lim }}}x_{n}\end{cases}}$

例2.5.1

（1）

（2）

（3）

（4）

定理2.5.2

（1）

（2）

（3）

定理2.5.3

（1）

（2）

（3）

（4）

例2.5.4

习题

习题二

1. 将有理数集排成序列

2. 能这样定义极限吗？

（1）不可。${\underset {n>1,\,n\in \mathbb {N} ^{*}}{\forall }}\left|{\frac {1}{10^{1000}}}\right|<{\frac {1}{10^{100}}}\nRightarrow \lim _{n\to \infty }{\frac {1}{10^{1000}}}=0$

（2）可。${\underset {\varepsilon >0}{\forall }}\ {\underset {k\in \mathbb {N} ^{*}}{\exists }}{\frac {1}{k}}\leqslant \varepsilon$

（3）可。${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} ,\,a\in \mathbb {R} ,\,M>0}{\forall }}\color {CadetBlue}\left(\color {black}{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<M\varepsilon \Leftrightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|x_{n}-a|<M{\frac {\varepsilon }{M}}=\varepsilon \right)$

3. 用定义证明序列极限

（1）${\mathit {true}}\Leftrightarrow {\underset {\varepsilon >0,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}\left|{\frac {\cos n}{n}}\right|<\varepsilon \Leftrightarrow {\frac {\left|\cos n\right|}{n}}<\varepsilon \Leftarrow {\frac {1}{n}}<\varepsilon \Leftrightarrow n>{\frac {1}{\varepsilon }}\right)\color {black}\Rightarrow \lim _{n\to \infty }{\frac {\cos n}{n}}=0$

（2）

（3）${\mathit {true}}\Leftrightarrow {\underset {\varepsilon >0,\,n\in \mathbb {N} ^{*}}{\forall }}\color {CadetBlue}\left(\color {black}|{\sqrt {n+1}}-{\sqrt {n}}|<\varepsilon \Leftrightarrow {\frac {1}{{\sqrt {n+1}}+{\sqrt {n}}}}<\varepsilon \Leftarrow {\frac {1}{2{\sqrt {n}}}}<\varepsilon \Leftrightarrow n>{\frac {1}{4\varepsilon ^{2}}}\right)\color {black}\Rightarrow \lim _{n\to \infty }({\sqrt {n+1}}-{\sqrt {n}})=0$

（4）

（5）

（6）

4.

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} ,\,\langle y_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} ,\,a\in \mathbb {R} }{\forall }}{\Big (}{\underset {n\in \mathbb {N} ^{*}}{\forall }}x_{n}\leqslant a\leqslant y_{n}\land \lim _{n\to \infty }(y_{n}-x_{n})=0$

$\Rightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}(0\leqslant a-x_{n}\leqslant y_{n}-x_{n}<\varepsilon \land 0\leqslant y_{n}-a\leqslant y_{n}-x_{n}<\varepsilon )\Rightarrow \lim _{n\to \infty }x_{n}=a=\lim _{n\to \infty }y_{n}{\Big )}$

5.

6. 用定义验证无穷大量

（1）

（2）

（3）Fibonacci 序列

7.

8. 用肯定的语气叙述序列不是无穷小量

${\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}x_{n}\neq o(1)(n\to \infty )\Leftrightarrow {\underset {\varepsilon >0}{\exists }}\ {\underset {N\in \mathbb {N} ^{*}}{\forall }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\exists }}|x_{n}|\geqslant \varepsilon \right)$

9. 余弦序列发散

10. 求序列极限

（1）

（2）

第三章 函数的极限与连续性

3.1 函数的极限

3.1.1 函数极限的定义

定义3.1.1：函数极限^w

${\begin{aligned}{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\underset {x_{0}\in \mathbb {R} ,\,A\in \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}\left(\color {black}{\underset {x\to x_{0}}{\lim }}f(x)=A\,\Leftrightarrow \,f(x)\xrightarrow {x\to x_{0}} A\ :\Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)-A|<\varepsilon \right)$

例3.1.1

${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {x\in U_{0}(1,{\color {RoyalBlue}1})}{\forall }}\left|{\frac {{\sqrt {x}}-1}{x-1}}-{\frac {1}{2}}\right|=\left|{\frac {1}{{\sqrt {x}}+1}}-{\frac {1}{2}}\right|=\left|{\frac {2-{\sqrt {x}}-1}{2({\sqrt {x}}+1)}}\right|=\left|{\frac {{\sqrt {x}}-1}{2({\sqrt {x}}+1)}}\right|={\frac {|x-1|}{2({\sqrt {x}}+1)^{2}}}<{\frac {|x-1|}{2}}\\&\Rightarrow {\underset {\varepsilon >0,\,x\in U_{0}(1,{\color {RoyalBlue}\min\{{\color {black}1},2\varepsilon \}})}{\forall }}\left|{\frac {{\sqrt {x}}-1}{x-1}}-{\frac {1}{2}}\right|<{\frac {\color {RoyalBlue}2\varepsilon }{2}}=\varepsilon \Rightarrow \color {darkgreen}\lim _{x\rightarrow 1}{\frac {{\sqrt {x}}-1}{x-1}}={\frac {1}{2}}\end{aligned}}$

例3.1.2

（1）${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {x\in U_{0}(2,{\color {RoyalBlue}1})}{\forall }}|x^{4}-16|=|x-2||x+2||x^{2}+4|<|x-2||{\color {RoyalBlue}3}+2||{\color {RoyalBlue}3}^{2}+4|=65|x-2|\\&\Rightarrow {\underset {\varepsilon >0,\,x\in U_{0}\left(2,{\color {RoyalBlue}\min \left\{{\color {black}1},{\frac {\varepsilon }{65}}\right\}}\right)}{\forall }}|x^{4}-16|<65\cdot {\color {RoyalBlue}{\frac {\varepsilon }{65}}}=\varepsilon \Rightarrow \color {darkgreen}\lim _{x\rightarrow 2}x^{4}=16\end{aligned}}$

（2）${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {a\in \mathbb {R} ,\,n\in \mathbb {N} ^{*},\,x\in \mathbb {R} }{\forall }}(x-a)\left(\sum _{j=0}^{n-1}a^{j}x^{n-1-j}\right)=\sum _{j=0}^{n-1}a^{j}x^{\color {RoyalBlue}n-j}-\sum _{j=0}^{n-1}a^{j\color {RoyalBlue}+1}x^{n-1-j}=\sum _{j=0}^{n-1}a^{j}x^{n-j}-\sum _{j=\color {RoyalBlue}1}^{\color {RoyalBlue}n}a^{j}x^{\color {RoyalBlue}n-j}=x^{n}-a^{n}\\&\Rightarrow {\underset {a\in \mathbb {R} ,\,n\in \mathbb {N} ^{*},\,x\in U_{0}(a,{\color {RoyalBlue}1})}{\forall }}|x^{n}-a^{n}|=|x-a|\sum _{j=0}^{n-1}|a|^{j}|x|^{n-1-j}<|x-a|\sum _{j=0}^{n-1}|a|^{j}({\color {RoyalBlue}|a|+1})^{n-1-j}\\&\Rightarrow {\underset {a\in \mathbb {R} ,\,n\in \mathbb {N} ^{*},\,\varepsilon >0,\,x\in U_{0}\left(a,\color {RoyalBlue}\min \left\{{\color {black}1},{\frac {\varepsilon }{\sum _{j=0}^{n-1}|a|^{j}(|a|+1)^{n-1-j}}}\right\}\right)}{\forall }}|x^{n}-a^{n}|<\varepsilon \Rightarrow \color {darkgreen}{\underset {a\in \mathbb {R} ,\,n\in \mathbb {N} ^{*}}{\forall }}\lim _{x\to a}x^{n}=a^{n}\end{aligned}}$（如遇 $0^{0}$ 则取 $1$）

例3.1.3

${\mathit {true}}\Leftrightarrow {\underset {\varepsilon >0,\,x\in U_{0}(0,{\color {RoyalBlue}\varepsilon })}{\forall }}\left|x\sin {\frac {1}{x}}\right|\leqslant |x|<\varepsilon \Rightarrow \color {darkgreen}\lim _{x\rightarrow 0}\left(x\sin {\frac {1}{x}}\right)=0$

3.1.2 函数极限的性质

定理3.1.1

（1）函数极限的唯一性

${\begin{aligned}{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\underset {A\in \mathbb {R} ,\,B\in \mathbb {R} ,\,\color {Sepia}A<B}{\underset {x_{0}\in \mathbb {R} }{\forall }}}}\end{aligned}}\color {CadetBlue}\left(\color {black}{\color {CadetBlue}{\begin{cases}\color {Sepia}\displaystyle \lim _{x\to x_{0}}f(x)=A\\\color {Sepia}\displaystyle \lim _{x\to x_{0}}f(x)=B\end{cases}}}\Rightarrow {\begin{aligned}{\underset {\delta _{2}>0}{\underset {\delta _{1}>0}{\exists }}}\end{aligned}}{\color {CadetBlue}{\begin{cases}\color {black}{\underset {x\in U_{0}(x_{0},\delta _{1})}{\forall }}|f(x)-A|<{\color {RoyalBlue}{\frac {B-A}{2}}}\\\color {black}{\underset {x\in U_{0}(x_{0},\delta _{2})}{\forall }}|f(x)-B|<\color {RoyalBlue}{\frac {B-A}{2}}\end{cases}}}\Rightarrow {\begin{array}{c}{\underset {\delta _{1}>0,\,\delta _{2}>0}{\exists }}\ {\underset {x\in U_{0}(x_{0},{\color {RoyalBlue}\min }\{\delta _{1},\delta _{2}\})}{\forall }}\\f(x)<{\frac {A+B}{2}}<f(x)\end{array}}\,\Leftrightarrow \,\color {darkgreen}{\mathit {false}}\right)$

（2）函数在取极限处周围的局部有界性

${\begin{aligned}{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\underset {x_{0}\in \mathbb {R} ,\,A\in \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}\left(\color {Sepia}\lim _{x\to x_{0}}f(x)=A\color {black}\Rightarrow {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)-A|<{\color {RoyalBlue}1}\Rightarrow \color {darkgreen}{\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)|<|A|+1\right)$

定理3.1.2：函数极限的保序性

（2）${\begin{aligned}{\underset {G\subseteq \mathbb {R} ,\,g:G\to \mathbb {R} }{\underset {F\subseteq \mathbb {R} ,\,f:F\to \mathbb {R} }{\underset {\color {Sepia}A<B,\,C\in (A,\,B)}{\underset {x_{0}\in \mathbb {R} ,\,A\in \mathbb {R} ,\,B\in \mathbb {R} }{\forall }}}}}\end{aligned}}\color {CadetBlue}\left({\begin{aligned}{\begin{cases}\color {Sepia}\displaystyle \lim _{x\to x_{0}}f(x)=A\\\color {Sepia}\displaystyle \lim _{x\to x_{0}}g(x)=B\end{cases}}&\color {black}{}\Rightarrow {\begin{aligned}{\underset {\delta _{2}>0}{\underset {\delta _{1}>0}{\exists }}}\end{aligned}}{\color {CadetBlue}{\begin{cases}\color {black}{\underset {x\in U_{0}(x_{0},\delta _{1})}{\forall }}|f(x)-A|<\color {RoyalBlue}C-A\\\color {black}{\underset {x\in U_{0}(x_{0},\delta _{2})}{\forall }}|g(x)-B|<\color {RoyalBlue}B-C\end{cases}}}\\&\color {black}{}\Rightarrow {\underset {\delta _{1}>0,\,\delta _{2}>0}{\exists }}\ {\underset {x\in U_{0}(x_{0},{\color {RoyalBlue}\min }\{\delta _{1},\delta _{2}\})}{\forall }}f(x)<C<g(x)\Leftrightarrow \color {darkgreen}{\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}f(x)<C<g(x)\end{aligned}}\right)$

（1）${\begin{aligned}{\underset {G\subseteq \mathbb {R} ,\,g:G\to \mathbb {R} }{\underset {F\subseteq \mathbb {R} ,\,f:F\to \mathbb {R} }{\underset {x_{0}\in \mathbb {R} ,\,A\in \mathbb {R} ,\,B\in \mathbb {R} }{\forall }}}}\end{aligned}}\color {CadetBlue}\left(\color {black}{\begin{aligned}&\color {Sepia}\lim _{x\to x_{0}}f(x)=A\ \land \,\lim _{x\to x_{0}}g(x)=B\ \land \ {\underset {\delta >0}{\forall }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\exists }}f(x)\leqslant g(x)\\\Rightarrow {}&{\color {CadetBlue}\left(\color {black}A>B\,\Rightarrow {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}g(x)<{\color {RoyalBlue}{\frac {A+B}{2}}}<f(x)\,\Leftrightarrow \,{\mathit {false}}\right)}\Rightarrow \color {darkgreen}A\leqslant B\end{aligned}}\right)$

定理3.1.3：函数极限的四则运算

（1）函数极限的加减法

${\begin{aligned}{\underset {x_{0}\in \mathbb {R} }{\underset {A\in \mathbb {R} ,\,B\in \mathbb {R} }{\underset {Y\subseteq \mathbb {R} ,\,g:Y\to \mathbb {R} }{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\forall }}}}}\end{aligned}}\color {CadetBlue}\left(\color {black}{\begin{aligned}&{\color {CadetBlue}{\begin{cases}\color {Sepia}\displaystyle \lim _{x\to x_{0}}f(x)=A\\\color {Sepia}\displaystyle \lim _{x\to x_{0}}g(x)=B\end{cases}}}\Leftrightarrow {\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {\delta _{1}>0,\,\delta _{2}>0}{\exists }}\end{array}}\color {CadetBlue}{\begin{cases}\color {black}{\underset {x\in U_{0}(x_{0},\delta _{1})}{\forall }}|f(x)-A|<{\color {RoyalBlue}{\frac {\varepsilon }{2}}}\\\color {black}{\underset {x\in U_{0}(x_{0},\delta _{2})}{\forall }}|g(x)-B|<\color {RoyalBlue}{\frac {\varepsilon }{2}}\end{cases}}\\\Rightarrow {}&{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta _{1}>0,\,\delta _{2}>0}{\exists }}\ {\underset {x\in U_{0}(x_{0},{\color {RoyalBlue}\min }\{\delta _{1},\delta _{2}\})}{\forall }}\\&|(f(x)\pm g(x))-(A\pm B)|=|(f(x)-A)\pm (g(x)-B)|\leqslant |f(x)-A|+|g(x)-B|<{\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2}}=\varepsilon \\\Rightarrow {}&\color {darkgreen}\lim _{x\to x_{0}}(f(x)\pm g(x))=A\pm B\end{aligned}}\right)$

（2）函数极限的乘法

${\begin{aligned}{\underset {x_{0}\in \mathbb {R} }{\underset {A\in \mathbb {R} ,\,B\in \mathbb {R} }{\underset {Y\subseteq \mathbb {R} ,\,g:Y\to \mathbb {R} }{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\forall }}}}}\end{aligned}}\color {CadetBlue}\left(\color {black}{\begin{aligned}&\color {CadetBlue}{\begin{cases}\color {Sepia}\displaystyle \lim _{x\to x_{0}}f(x)=A\\\color {Sepia}\displaystyle \lim _{x\to x_{0}}g(x)=B\end{cases}}\color {black}\Rightarrow {\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {\delta _{2}>0}{\underset {\delta _{1}>0}{\underset {\delta >0}{\exists }}}}\end{array}}\color {CadetBlue}{\begin{cases}\color {black}{\underset {x\in U_{0}(x_{0},\delta )}{\forall }}\ \,|g(x)|<|A|+1\\\color {black}{\underset {x\in U_{0}(x_{0},\delta _{1})}{\forall }}|f(x)-A|<\color {RoyalBlue}{\frac {\varepsilon }{2(|A|+1)}}\\\color {black}{\underset {x\in U_{0}(x_{0},\delta _{2})}{\forall }}|g(x)-B|<\color {RoyalBlue}{\frac {\varepsilon }{2(|A|+1)}}\end{cases}}\\\Rightarrow {}&{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0,\,\delta _{1}>0,\,\delta _{2}>0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\,{\color {RoyalBlue}\min }\{\delta ,\,\delta _{1},\,\delta _{2}\})}{\forall }}\\&|f(x)g(x)-AB|=|f(x)g(x){{}\color {RoyalBlue}-Ag(x)+Ag(x)}-AB|\leqslant |g(x)||f(x)-A|+|A||g(x)-B|\\&{\hphantom {|f(x)g(x)-AB|}}<(|A|+1){\frac {\varepsilon }{2(|A|+1)}}+|A|{\frac {\varepsilon }{2(|A|+1)}}<\varepsilon \\\Rightarrow {}&\color {darkgreen}\lim _{n\to \infty }(f(x)g(x))=AB\end{aligned}}\right)$

（3）函数极限的除法

${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\begin{aligned}{\underset {x_{0}\in \mathbb {R} }{\underset {B\in \mathbb {R} }{\underset {g:Y\to \mathbb {R} }{\underset {Y\subseteq \mathbb {R} }{\forall }}}}}\end{aligned}}\!\!\color {CadetBlue}\left(\color {black}{\begin{aligned}\!{\color {Sepia}{\begin{array}{c}\displaystyle \lim _{x\to x_{0}}g(x)\\=B\neq 0\end{array}}}\!&\Rightarrow \!{\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {\delta _{2}>0}{\underset {\delta _{1}>0}{\exists }}}\end{array}}\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}{\mathit {true}}&\Leftrightarrow {\underset {x\in U_{0}(x_{0},\delta _{1})}{\forall }}|g(x)|>{\color {RoyalBlue}{\frac {|B|}{2}}}\,\land \,{\underset {x\in U_{0}(x_{0},\delta _{2})}{\forall }}|g(x)-B|<{\color {RoyalBlue}{\frac {|B|^{2}\varepsilon }{2}}}\\&\Rightarrow {\underset {x\in U_{0}(x_{0},{\color {RoyalBlue}\min }\{\delta _{1},\delta _{2}\})}{\forall }}\left|{\frac {1}{g(x)}}-{\frac {1}{B}}\right|=\left|{\frac {g(x)-B}{g(x)B}}\right|={\frac {|g(x)-B|}{|g(x)||B|}}<{\frac {\frac {|B|^{2}\varepsilon }{2}}{{\frac {|B|}{2}}|B|}}=\varepsilon \end{aligned}}\!\right)\\&\Rightarrow \color {darkgreen}\lim _{x\to x_{0}}{\frac {1}{g(x)}}={\frac {1}{B}}\end{aligned}}\!\!\right)\\&\Rightarrow \!{\begin{aligned}{\underset {A\in \mathbb {R} ,\,B\in \mathbb {R} ,\,x_{0}\in \mathbb {R} }{\underset {Y\subseteq \mathbb {R} ,\,g:Y\to \mathbb {R} }{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\forall }}}}\end{aligned}}\!\color {CadetBlue}\left(\color {Sepia}\lim _{x\to x_{0}}f(x)=A\ \land \,\lim _{x\to x_{0}}g(x)=B\neq 0\color {black}\ \Rightarrow \,{\color {darkgreen}\lim _{x\to x_{0}}{\frac {f(x)}{g(x)}}}=\lim _{x\to x_{0}}f(x)\cdot \lim _{x\to x_{0}}{\frac {1}{g(x)}}\color {darkgreen}={\frac {A}{B}}\right)\end{aligned}}$

定理3.1.4：复合函数的极限

${\begin{aligned}{\underset {x_{0}\in \mathbb {R} ,\,\color {RoyalBlue}\delta _{0}>0}{\underset {A\in \mathbb {R} ,\,u_{0}\in \mathbb {R} }{\underset {g:X\to \mathbb {R} }{\underset {f:U\to \mathbb {R} }{\underset {U\subseteq \mathbb {R} ,\,X\subseteq \mathbb {R} }{\forall }}}}}}\end{aligned}}\!\!\color {CadetBlue}\left(\left\{\!\color {Sepia}{\begin{aligned}&\lim _{u\to u_{0}}f(u)=A\\&\lim _{x\to x_{0}}g(x)=u_{0}\\&{\underset {x\in U_{0}(x_{0},\delta _{0})}{\forall }}g(x)\color {RoyalBlue}\neq u_{0}\end{aligned}}\!\!\right\}{\color {black}\Rightarrow }\left\{\color {black}{\begin{array}{c}{\underset {\varepsilon >0{\vphantom {\eta }}}{\forall }}\\{\underset {\eta >0}{\forall }}\end{array}}{\begin{array}{c}{\underset {\eta >0}{\exists }}\\{\underset {\delta \in (0,\color {RoyalBlue}\delta _{0})}{\exists }}\end{array}}{\begin{array}{c}{\underset {u\in U_{0}(u_{0},\eta )}{\forall }}\\{\underset {x\in U_{0}(x_{0},\delta )}{\forall }}\end{array}}\!{\begin{array}{r}{\underset {}{|f(u)-\,A|<\varepsilon }}\\{\underset {}{{\color {RoyalBlue}0<{}}|g(x)-u_{0}|<\eta }}\end{array}}\right.\color {black}\Rightarrow \!{\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}\\|f(g(x))-A|<\varepsilon \end{array}}\!\Rightarrow \color {darkgreen}{\begin{array}{c}\displaystyle \lim _{x\to x_{0}}f(g(x))\\=A\end{array}}\!\!\right)$

定理3.1.5：函数的夹逼收敛原理

${\begin{aligned}{\underset {A\in \mathbb {R} ,\,x_{0}\in \mathbb {R} ,\,\delta _{0}>0}{\underset {Z\subseteq \mathbb {R} ,\,h:Z\to \mathbb {R} }{\underset {Y\subseteq \mathbb {R} ,\,g:Y\to \mathbb {R} }{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\forall }}}}}\end{aligned}}\!\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}&\color {Sepia}\lim _{x\to x_{0}}f(x)=\lim _{x\to x_{0}}h(x)=A\ \land \,{\underset {x\in U_{0}(x_{0},\delta _{0})}{\forall }}f(x)\leqslant g(x)\leqslant h(x)\\\Rightarrow {}&{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta _{1}>0,\,\delta _{2}>0}{\exists }}\color {CadetBlue}\left(\color {black}{\underset {x\in U_{0}(x_{0},\delta _{0})}{\forall }}f(x)\leqslant g(x)\leqslant h(x)\land {\underset {x\in U_{0}(x_{0},\delta _{1})}{\forall }}|f(x)-A|<\varepsilon \land {\underset {x\in U_{0}(x_{0},\delta _{2})}{\forall }}|h(x)-A|<\varepsilon \right)\\\Rightarrow {}&{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta _{1}>0,\,\delta _{2}>0}{\exists }}\ {\underset {x\in U_{0}(x_{0},{\color {RoyalBlue}\min }\{\delta _{0},\delta _{1},\delta _{2}\})}{\forall }}{\color {CadetBlue}(}{\mathit {true}}\,\Leftrightarrow \,A-\varepsilon <f(x)\leqslant g(x)\leqslant h(x)<A+\varepsilon \,\Rightarrow \,|g(x)-A|<\varepsilon {\color {CadetBlue})}\\\Rightarrow {}&\color {darkgreen}\lim _{x\to x_{0}}g(x)=A\end{aligned}}\!\!\right)$

3.1.3 函数极限概念的推广

定义3.1.2：单侧极限（右极限、左极限）

${\begin{aligned}{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\underset {x_{0}\in \mathbb {R} ,\,A\in \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}{\begin{cases}\!\color {black}\displaystyle \lim _{x\to x_{0}\color {RoyalBlue}+0}f(x)=A\Leftrightarrow f(x_{0}{\color {RoyalBlue}{}+0})=A\,:\Leftrightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}^{\color {RoyalBlue}+}(x_{0},\delta )}{\forall }}|f(x)-A|<\varepsilon \\\!\color {black}\displaystyle \lim _{x\to x_{0}\color {RoyalBlue}-0}f(x)=A\Leftrightarrow f(x_{0}{\color {RoyalBlue}{}-0})=A\,:\Leftrightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}^{\color {RoyalBlue}-}(x_{0},\delta )}{\forall }}|f(x)-A|<\varepsilon \end{cases}}$

定理3.1.6：函数极限存在的充要条件

${\begin{aligned}{\mathit {true}}&\Leftrightarrow {\begin{aligned}{\underset {x_{0}\in \mathbb {R} }{\underset {A\in \mathbb {R} }{\underset {f:X\to \mathbb {R} }{\underset {X\subseteq \mathbb {R} }{\forall }}}}}\end{aligned}}\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}&\lim _{x\to x_{0}}f(x)=A\ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)-A|<\varepsilon \ \Rightarrow \ f(x_{0}+0)=f(x_{0}-0)=A\\\Leftrightarrow {}&{\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {\delta _{1}>0,\,\delta _{2}>0}{\exists }}\end{array}}{\color {CadetBlue}{\begin{cases}\color {black}{\underset {x\in U_{0}^{+}(x_{0},\delta _{1})}{\forall }}|f(x)-A|<\varepsilon \\\color {black}{\underset {x\in U_{0}^{-}(x_{0},\delta _{2})}{\forall }}|f(x)-A|<\varepsilon \end{cases}}}\Rightarrow {\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta _{1}>0,\,\delta _{2}>0}{\exists }}\ {\underset {x\in U_{0}(x_{0},{\color {RoyalBlue}\min }\{\delta _{1},\delta _{2}\})}{\forall }}\\|f(x)-A|<\varepsilon \end{array}}\Rightarrow {\begin{array}{c}\displaystyle \lim _{x\to x_{0}}f(x)\\=A\end{array}}\end{aligned}}\!\!\right)\\&\Rightarrow {\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} ,\,A\in \mathbb {R} ,\,x_{0}\in \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {darkgreen}\lim _{x\to x_{0}}f(x)=A\ \Leftrightarrow \ f(x_{0}+0)=f(x_{0}-0)=A\right)\end{aligned}}$

定义3.1.3：自变量趋向（正、负）无穷大时的极限

${\begin{aligned}{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\underset {A\in \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}{\begin{cases}\!\color {black}\displaystyle \lim _{x\to +\infty }f(x)=A\Leftrightarrow f(x)\xrightarrow {x\to +\infty } A\,:\Leftrightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {h\in \mathbb {R} }{\exists }}\ \,{\underset {x>h}{\forall }}\ |f(x)-A|<\varepsilon \\\!\color {black}\displaystyle \lim _{x\to -\infty }f(x)=A\Leftrightarrow f(x)\xrightarrow {x\to -\infty } A\,:\Leftrightarrow {\underset {\varepsilon >0}{\forall }}\ {\underset {h\in \mathbb {R} }{\exists }}\ \,{\underset {x<h}{\forall }}\ |f(x)-A|<\varepsilon \\\!\color {black}\displaystyle {\overset {\hphantom {x\to +\infty }}{\lim _{x\to \infty }}}\,f(x)=A\Leftrightarrow f(x)\xrightarrow {\ x\to \infty \ } A\,:\Leftrightarrow \!{\begin{aligned}{\underset {\varepsilon >0}{\forall }}\ {\underset {h\in \mathbb {R} }{\exists }}\end{aligned}}\!{\begin{aligned}{\underset {|x|>h\,}{\underset {x\in \mathbb {R} }{\forall }}}\end{aligned}}\!|f(x)-A|<\varepsilon \end{cases}}$

定义3.1.4：广义极限

${\begin{aligned}{\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} }{\underset {x_{0}\in \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}{\begin{cases}\!\color {black}\displaystyle \lim _{x\to x_{0}}f(x)=+\infty \Leftrightarrow f(x)\xrightarrow {x\to x_{0}} +\infty \,:\Leftrightarrow {\underset {M>0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}f(x)>M\\\!\color {black}\displaystyle \lim _{x\to x_{0}}f(x)=-\infty \Leftrightarrow f(x)\xrightarrow {x\to x_{0}} -\infty \,:\Leftrightarrow {\underset {M>0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}f(x)<-M\\\!\color {black}\displaystyle \lim _{x\to x_{0}}f(x)={\phantom {+}}\infty \Leftrightarrow f(x)\xrightarrow {x\to x_{0}} {\phantom {+}}\infty \,:\Leftrightarrow {\underset {M>0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)|>M\end{cases}}$

$x\to x_{0},\ x_{0}\pm 0,\,\pm \infty ,\ \infty ;\quad f(x)\to A,\,\pm \infty ,\ \infty {\text{.}}\quad$函数极限共 24 种情况。

3.1.4 序列极限与函数极限的关系

${\begin{aligned}{\underset {{\color {Sepia}U_{0}(x_{0},\delta _{0})\subseteq X},\,f:X\to \mathbb {R} }{\underset {x_{0}\in \mathbb {R} ,\,A\in \mathbb {R} ,\,{\color {Sepia}\delta _{0}>0},\,X\subseteq \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}\left(\color {darkgreen}{\underset {x\to x_{0}}{\lim }}f(x){\color {RoyalBlue}{}\neq {}}A\,\color {black}\Leftrightarrow \,{\color {RoyalBlue}\neg }\ {\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)-A|<\varepsilon \,\color {darkgreen}\Leftrightarrow {\underset {\varepsilon >0}{\color {RoyalBlue}\exists }}\ {\underset {\delta >0}{\color {RoyalBlue}\forall }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\color {RoyalBlue}\exists }}|f(x)-A|\geqslant \varepsilon \right)$

定理3.1.7：序列极限与函数极限的关系

${\begin{aligned}{\mathit {true}}&\Leftrightarrow \!{\begin{aligned}{\underset {f:X\to \mathbb {R} }{\underset {U_{0}(x_{0},\delta _{0})\subseteq X}{\underset {\delta _{0}>0,\,X\subseteq \mathbb {R} }{\underset {A\in \mathbb {R} ,\,x_{0}\in \mathbb {R} }{\forall }}}}}\end{aligned}}\color {CadetBlue}{\begin{cases}\color {black}\!\displaystyle \lim _{x\to x_{0}}f(x)=A\,\Rightarrow {\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\color {CadetBlue}\!\left(\color {black}{\begin{aligned}&\color {CadetBlue}{\begin{cases}\color {black}{\underset {n\in \mathbb {N} ^{*}}{\forall }}x_{n}\in U_{0}(x_{0},\delta _{0})\\\color {black}\displaystyle \lim _{n\to \infty }x_{n}=x_{0}\end{cases}}{\color {black}\Rightarrow {}}{\begin{cases}\color {black}{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)-A|<\varepsilon \\\color {black}{\underset {\delta >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}\in U_{0}(x_{0},\delta )\end{cases}}\\\color {black}\Rightarrow \,&\,\color {black}{\underset {\varepsilon >0}{\forall }}\ {\underset {n\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}|f(x_{n})-A|<\varepsilon \ \Leftrightarrow \,\lim _{n\to \infty }f(x_{n})=A\end{aligned}}\!\right)\\\color {black}\!\!{\begin{aligned}\lim _{x\to x_{0}}f(x)\neq A\,&\Rightarrow {\underset {\varepsilon >0}{\exists }}\ {\underset {\color {RoyalBlue}n\in \mathbb {N} ^{*}}{\forall }}\ {\underset {x\in U_{0}(x_{0},\,{\color {RoyalBlue}\delta _{0}/n})}{\exists }}|f(x)-A|\geqslant \varepsilon \\&\Rightarrow {\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\exists }}\color {CadetBlue}{\begin{cases}\color {black}{\mathit {true}}\Leftrightarrow {\underset {n\in \mathbb {N} ^{*}}{\forall }}{\color {RoyalBlue}x_{n}}\!\in U_{0}\!\left(x_{0},{\frac {\delta _{0}}{n}}\right)\!\subseteq U_{0}(x_{0},\delta _{0})\,\Rightarrow \displaystyle \lim _{n\to \infty }x_{n}=x_{0}\\\color {black}{\mathit {true}}\Leftrightarrow {\underset {\varepsilon >0}{\exists }}\ {\underset {n\in \mathbb {N} ^{*}}{\forall }}|f({\color {RoyalBlue}x_{n}})-A|\geqslant \varepsilon \,\Rightarrow \displaystyle \lim _{n\to \infty }f(x_{n})\neq A\end{cases}}\end{aligned}}\end{cases}}\\&\Rightarrow \!{\begin{aligned}{\underset {f:X\to \mathbb {R} }{\underset {\color {Sepia}U_{0}(x_{0},\delta _{0})\subseteq X}{\underset {{\color {Sepia}\delta _{0}>0},\,X\subseteq \mathbb {R} }{\underset {A\in \mathbb {R} ,\,x_{0}\in \mathbb {R} }{\forall }}}}}\end{aligned}}\!\color {CadetBlue}\left(\color {darkgreen}\lim _{x\to x_{0}}f(x)=A\,\Leftrightarrow {\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\!\color {CadetBlue}\left({\begin{cases}\color {darkgreen}{\underset {n\in \mathbb {N} ^{*}}{\forall }}x_{n}\in U_{0}(x_{0},\delta _{0})\\\color {darkgreen}\displaystyle \lim _{n\to \infty }x_{n}=x_{0}\end{cases}}\!\color {darkgreen}\Rightarrow \lim _{n\to \infty }f(x_{n})=A\right)\!\right)\end{aligned}}$

3.1.5 极限存在性定理和两个重要极限

定理3.1.8（极限存在性定理）：单调函数必有广义极限

${\begin{aligned}{\underset {f:X\to \mathbb {R} }{\underset {X\subseteq \mathbb {R} }{\underset {\delta _{0}>0}{\underset {x_{0}\in \mathbb {R} }{\forall }}}}}\end{aligned}}\color {CadetBlue}{\begin{cases}\!\!\!\color {Sepia}{\begin{aligned}{\begin{array}{c}{\underset {x_{1}<x_{2}}{\underset {x_{2}\in U_{0}^{+}(x_{0},\delta _{0})}{\underset {x_{1}\in U_{0}^{+}(x_{0},\delta _{0})}{\forall }}}}\\f(x_{1})\leqslant f(x_{2})\end{array}}&\color {black}\Rightarrow \color {CadetBlue}{\begin{cases}\!{\color {black}{\underset {A\in \mathbb {R} }{\forall }}}\left(\color {black}{\begin{aligned}\!\!{\begin{array}{c}\displaystyle \inf _{x\in U_{0}^{+}(x_{0},\delta _{0})}f(x)\\=A\end{array}}\!&\Rightarrow \!{\begin{array}{c}{\underset {\varepsilon >0}{\forall }}\\{\underset {x_{1}\in U_{0}^{+}(x_{0},\delta _{0})}{\exists }}\end{array}}\!\!\color {CadetBlue}\left(\color {black}{\begin{aligned}{\mathit {true}}&\Leftrightarrow A\leqslant f(x_{1})<A+\varepsilon \\&\Leftrightarrow {\underset {x\in (x_{0},\,x_{1})}{\forall }}A\leqslant f(x)\leqslant f(x_{1})<A+\varepsilon \end{aligned}}\!\right)\\&\Rightarrow \lim _{x\to x_{0}+0}f(x)=A\end{aligned}}\!\!\right)\\\!\color {black}\displaystyle {\begin{array}{c}\displaystyle \inf _{x\in U_{0}^{+}(x_{0},\delta _{0})}f(x)\\=-\infty \end{array}}\!\Rightarrow \!{\begin{array}{c}{\underset {M>0}{\forall }}\\{\underset {x_{1}\in U_{0}(x_{0},\delta _{0})}{\exists }}\end{array}}\!\!\color {CadetBlue}\left(\color {black}{\begin{aligned}{\mathit {true}}&\Leftrightarrow f(x_{1})<-M\\&\Leftrightarrow {\underset {x\in (x_{0},\,x_{1})}{\forall }}f(x)\leqslant f(x_{1})<-M\end{aligned}}\!\right)\color {black}\Rightarrow {\begin{array}{c}\displaystyle \lim _{x\to x_{0}+0}f(x)\\=-\infty \end{array}}\end{cases}}\\&\color {black}\Rightarrow \color {darkgreen}\lim _{x\to x_{0}+0}f(x)=\inf _{x\in U_{0}^{+}(x_{0},\delta _{0})}f(x)\end{aligned}}\\\!\!\color {Sepia}{\begin{aligned}&{\underset {x_{1}\in U_{0}^{+}(x_{0},\delta _{0}),\,x_{2}\in U_{0}^{+}(x_{0},\delta _{0}),\,x_{1}<x_{2}}{\forall }}f(x_{1})\geqslant f(x_{2})\ \color {black}\Rightarrow \,\color {darkgreen}\lim _{x\to x_{0}+0}f(x)=\sup _{x\in U_{0}^{+}(x_{0},\delta _{0})}f(x)\\&{\underset {x_{1}\in U_{0}^{-}(x_{0},\delta _{0}),\,x_{2}\in U_{0}^{-}(x_{0},\delta _{0}),\,x_{1}<x_{2}}{\forall }}f(x_{1})\leqslant f(x_{2})\ \color {black}\Rightarrow \,\color {darkgreen}\lim _{x\to x_{0}-0}f(x)=\sup _{x\in U_{0}^{-}(x_{0},\delta _{0})}f(x)\\&{\underset {x_{1}\in U_{0}^{-}(x_{0},\delta _{0}),\,x_{2}\in U_{0}^{-}(x_{0},\delta _{0}),\,x_{1}<x_{2}}{\forall }}f(x_{1})\geqslant f(x_{2})\ \color {black}\Rightarrow \,\color {darkgreen}\lim _{x\to x_{0}-0}f(x)=\inf _{x\in U_{0}^{-}(x_{0},\delta _{0})}f(x)\end{aligned}}\end{cases}}$

${\begin{aligned}{\underset {f:X\to \mathbb {R} }{\underset {X\subseteq \mathbb {R} }{\underset {x_{0}\in \mathbb {R} }{\forall }}}}\end{aligned}}\color {CadetBlue}{\begin{cases}\!\!\color {Sepia}{\begin{aligned}&{\underset {x_{1}\in \mathbb {R} ,\,x_{2}\in \mathbb {R} ,\,x_{0}<x_{1}<x_{2}}{\forall }}f(x_{1})\leqslant f(x_{2})\ \color {black}\Rightarrow \,\color {darkgreen}\lim _{x\to +\infty }f(x)=\sup _{x>x_{0}}f(x)\\&{\underset {x_{1}\in \mathbb {R} ,\,x_{2}\in \mathbb {R} ,\,x_{0}<x_{1}<x_{2}}{\forall }}f(x_{1})\geqslant f(x_{2})\ \color {black}\Rightarrow \,\color {darkgreen}\lim _{x\to +\infty }f(x)=\inf _{x>x_{0}}f(x)\\&{\underset {x_{1}\in \mathbb {R} ,\,x_{2}\in \mathbb {R} ,\,x_{1}<x_{2}<x_{0}}{\forall }}f(x_{1})\leqslant f(x_{2})\ \color {black}\Rightarrow \,\color {darkgreen}\lim _{x\to -\infty }f(x)=\inf _{x<x_{0}}f(x)\\&{\underset {x_{1}\in \mathbb {R} ,\,x_{2}\in \mathbb {R} ,\,x_{1}<x_{2}<x_{0}}{\forall }}f(x_{1})\geqslant f(x_{2})\ \color {black}\Rightarrow \,\color {darkgreen}\lim _{x\to -\infty }f(x)=\sup _{x<x_{0}}f(x)\end{aligned}}\end{cases}}$

定理3.1.9（函数的 Cauchy 收敛准则）

${\begin{aligned}{\mathit {true}}&\Leftrightarrow \!{\begin{aligned}{\underset {f:X\to \mathbb {R} }{\underset {\color {Sepia}U_{0}(x_{0},\delta _{0})\subseteq X}{\underset {{\color {Sepia}\delta _{0}>0},\,X\subseteq \mathbb {R} }{\underset {x_{0}\in \mathbb {R} }{\forall }}}}}\end{aligned}}\!\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}&{\underset {A\in \mathbb {R} }{\exists }}\lim _{x\to x_{0}}f(x)=A\ \Leftrightarrow \,{\underset {A\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)-A|<\color {RoyalBlue}{\frac {\varepsilon }{2}}\\\Rightarrow \,&{\underset {A\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x_{1}\in U_{0}(x_{0},\delta ),\,x_{2}\in U_{0}(x_{0},\delta )}{\forall }}|f(x_{1})-f(x_{2})|\leqslant |f(x_{1})-A|-|f(x_{2})-A|<{\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2}}=\varepsilon \\\Rightarrow \,&\,{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x_{1}\in U_{0}(x_{0},\delta ),\,x_{2}\in U_{0}(x_{0},\delta )}{\forall }}|f(x_{1})-f(x_{2})|<\varepsilon \\\Rightarrow \,&{\underset {\langle x_{n}\rangle :\mathbb {N} ^{*}\to \mathbb {R} }{\forall }}\!\color {CadetBlue}\left(\!\color {black}{\begin{aligned}&{\underset {n\in \mathbb {N} ^{*}}{\forall }}x_{n}\in U_{0}(x_{0},\delta _{0})\ \land \,\lim _{n\to \infty }x_{n}=x_{0}\ \Rightarrow \,{\underset {\delta >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N,\,n\in \mathbb {N} ^{*}}{\forall }}x_{n}\in U_{0}(x_{0},\delta )\\\Rightarrow \,&\color {CadetBlue}{\begin{cases}\!\color {black}\displaystyle {\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0,\,N_{1}\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N_{1},\,n\in \mathbb {N} ^{*},\,x\in U_{0}(x_{0},\delta )}{\forall }}|f(x)-f(x_{n})|<\color {RoyalBlue}{\frac {\varepsilon }{2}}\\\!\color {black}{\mathit {true}}\Leftrightarrow \!{\begin{aligned}{\underset {\varepsilon >0}{\forall }}\ {\underset {N\in \mathbb {N} ^{*}}{\exists }}\end{aligned}}\!{\begin{aligned}{\underset {n>N,\,n\in \mathbb {N} ^{*}}{\underset {m>N,\,m\in \mathbb {N} ^{*}}{\forall }}}\end{aligned}}|f(x_{m})-f(x_{n})|<\varepsilon \Leftrightarrow \langle f(x_{n})\rangle _{n\in \mathbb {N} ^{*}}{\text{ is Cauchy}}\\\!\color {black}\displaystyle {\phantom {\mathit {true}}}\Leftrightarrow \!{\underset {A\in \mathbb {R} }{\exists }}\lim _{n\to \infty }f(x_{n})=A\,\Rightarrow {\underset {A\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {N_{2}\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>N_{2},\,n\in \mathbb {N} ^{*}}{\forall }}|f(x_{n})-A|<\color {RoyalBlue}{\frac {\varepsilon }{2}}\end{cases}}\\\Rightarrow \,&{\begin{array}{c}{\underset {A\in \mathbb {R} }{\exists }}\ {\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0,\,N_{1}\in \mathbb {N} ^{*},\,N_{2}\in \mathbb {N} ^{*}}{\exists }}\ {\underset {n>{\color {RoyalBlue}\max }\{N_{1},N_{2}\},\,n\in \mathbb {N} ^{*},\,x\in U_{0}(x_{0},\delta )}{\forall }}\\|f(x)-A|\leqslant |f(x)-f(x_{n})|+|f(x_{n})-A|<{\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2}}=\varepsilon \end{array}}\Rightarrow {\begin{array}{c}\exists _{A\in \mathbb {R} }\\\displaystyle \lim _{x\to x_{0}}f(x)=A\end{array}}\end{aligned}}\!\!\right)\end{aligned}}\!\!\right)\\&\Rightarrow {\underset {x_{0}\in \mathbb {R} ,\,{\color {Sepia}\delta _{0}>0},\,X\subseteq \mathbb {R} ,\,{\color {Sepia}U_{0}(x_{0},\delta _{0})\subseteq X},\,f:X\to \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {darkgreen}{\underset {A\in \mathbb {R} }{\exists }}\lim _{x\to x_{0}}f(x)=A\ \Leftrightarrow \,{\underset {\varepsilon >0}{\forall }}\ {\underset {\delta >0}{\exists }}\ {\underset {x_{1}\in U_{0}(x_{0},\delta ),\,x_{2}\in U_{0}(x_{0},\delta )}{\forall }}|f(x_{1})-f(x_{2})|<\varepsilon \right)\end{aligned}}$

第一个重要极限

使用：单位圆，三角形和扇形的面积，三角函数的奇偶性，差化积公式，夹逼收敛原理

${\mathit {true}}\Leftrightarrow {\color {RoyalBlue}{\underset {x\in (0,\,\pi /2)}{\forall }}{\frac {\sin x}{2}}<{\frac {x}{2}}<{\frac {\tan x}{2}}}\Leftrightarrow \color {darkgreen}{\underset {x\in (0,\,\pi /2)}{\forall }}\sin x<x<\tan x$

${\begin{aligned}{\mathit {true}}&\Leftrightarrow \color {CadetBlue}\left\{{\begin{array}{l}\color {black}\sin {\color {RoyalBlue}0}=\color {RoyalBlue}0\\\color {black}{\underset {x\in (0,\,\pi /2)}{\forall }}0<\sin x<x\\\color {black}{\underset {x\geqslant \pi /2}{\forall }}|{\sin x}|\leqslant 1<{\frac {\pi }{2}}\leqslant x\end{array}}\!\right\}\color {black}\Rightarrow {\begin{array}{c}\forall _{x\geqslant 0}\\|{\sin x}|=|{\sin({\color {RoyalBlue}-x})}|\leqslant |x|=|{\color {RoyalBlue}-x}|\end{array}}\Rightarrow {\begin{array}{c}\forall _{x\color {RoyalBlue}\in \mathbb {R} }\\|{\sin x}|\leqslant |x|\end{array}}\\&\Rightarrow {\begin{aligned}{\underset {x\in \mathbb {R} }{\underset {a\in \mathbb {R} }{\forall }}}\end{aligned}}\color {CadetBlue}\left\{\!\color {black}{\begin{array}{l}|{\cos x-\cos a}|=\left|2\sin {\frac {x+a}{2}}\sin {\frac {x-a}{2}}\right|\\|{\sin x\,-\sin a}|=\left|2\cos {\frac {x+a}{2}}\sin {\frac {x-a}{2}}\right|\end{array}}\!\leqslant 2\left|{\color {RoyalBlue}\sin {\frac {x-a}{2}}}\right|=2\left|{\color {RoyalBlue}{\frac {x-a}{2}}}\right|=|x-a|\right.\color {black}\Rightarrow {\underset {a\in \mathbb {R} }{\forall }}\color {CadetBlue}{\begin{cases}\color {darkgreen}\displaystyle \lim _{x\to a}\cos x=\cos a\\\color {darkgreen}\displaystyle \lim _{x\to a}\sin x=\sin a\end{cases}}\end{aligned}}$

${\mathit {true}}\Leftrightarrow {\color {CadetBlue}\left\{\!\color {black}{\begin{aligned}&{\mathit {true}}\Leftrightarrow {\underset {x\in (0,\,\pi /2)}{\forall }}0<\sin x<x<\tan x\,\Rightarrow {\underset {x\in (0,\pi /2)}{\forall }}\cos x={\color {RoyalBlue}{\frac {\sin x}{\color {black}\tan x}}}<{\color {RoyalBlue}{\frac {\sin x}{\color {black}x}}}<{\color {RoyalBlue}{\frac {\sin x}{\color {black}\sin x}}}=1\\&{\phantom {\mathit {true}}}\Leftrightarrow {\underset {x\in (0,\,\pi /2)}{\forall }}\cos x=\cos({\color {RoyalBlue}-x})<{\frac {\sin x}{x}}={\frac {\sin({\color {RoyalBlue}-x})}{\color {RoyalBlue}-x}}<1\,\Rightarrow {\underset {x\in \color {RoyalBlue}U_{0}(x,\,\pi /2)}{\forall }}\cos x<{\frac {\sin x}{x}}<1\\&\lim _{x\to \color {RoyalBlue}0}\cos x=\cos {\color {RoyalBlue}0}=1\end{aligned}}\!\right\}}\Rightarrow \color {darkgreen}{\begin{array}{c}\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}\\=1\end{array}}$

第四章 导数与微分

4.1 导数

4.1.2 导数的定义

定义 4.1.1：可导和导数

${\underset {X\subseteq \mathbb {R} ,\,f:X\to \mathbb {R} ,\,x\in \mathbb {R} }{\forall }}\color {CadetBlue}\left(\color {black}f{\text{ is differentiable at }}x:\Leftrightarrow {\underset {A\in \mathbb {R} }{\exists }}\,\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}=A=:{df \over dx}=f'(x)\right)$

例 4.1.1：常值函数的导数

${\underset {C\in \mathbb {R} ,\,x\in \mathbb {R} }{\forall }}{\color {darkgreen}{dC \over dx}}=\lim _{h\to 0}{\frac {C-C}{h}}\color {darkgreen}=0$

例 4.2.2

${\underset {x>0}{\forall }}{\color {darkgreen}{d{\sqrt {x}} \over dx}}=\lim _{h\rightarrow 0}{\frac {{\sqrt {x+h}}-{\sqrt {x}}}{h}}=\lim _{h\rightarrow 0}{\frac {1}{{\sqrt {x+h}}+{\sqrt {x}}}}\color {darkgreen}={\frac {1}{2{\sqrt {x}}}}$