# 極限 (數列)

## 定義

「对于任意的正实数 ${\displaystyle \epsilon >0}$，存在自然数 ${\displaystyle n\in \mathbb {N} }$ ，使得任意的自然数 ${\displaystyle i\in \mathbb {N} }$，只要 ${\displaystyle i>n}$，則 ${\displaystyle |z_{i}-z|<\epsilon }$

${\displaystyle (\forall \epsilon >0)(\exists n\in \mathbb {N} )(\forall n\in \mathbb {N} )[\,(i>n)\Rightarrow (|z_{i}-z|<\epsilon )\,]}$

${\displaystyle \lim _{i\to \infty }z_{i}=z}$

### 實數數列的極限

${\displaystyle \lim _{i\to \infty }z_{i}=z}$

${\displaystyle {\sqrt {{(z_{i})}^{2}+{(\operatorname {Im} (z))}^{2}}}<|\operatorname {Im} (z)|}$

## 基本性質

### 唯一性

${\displaystyle |z_{i}-z_{1}|<{\frac {\epsilon }{2}}}$
${\displaystyle |z_{i}-z_{2}|<{\frac {\epsilon }{2}}}$

${\displaystyle |z_{1}-z_{2}|=|(z_{1}-z_{i})-(z_{2}-z_{i})|\leq |z_{1}-z_{i}|+|z_{2}-z_{i}|<\epsilon }$

${\displaystyle |z_{1}-z_{2}|<|z_{1}-z_{2}|}$

### 有界性

（即 ${\displaystyle \{x_{i}\in \mathbb {R} \}_{i\in \mathbb {N} }}$ 有極限則必為有界數列）

${\displaystyle \lim _{n\to \infty }x_{n}=L}$

${\displaystyle |x_{i}-L|<\epsilon =1}$

${\displaystyle |x_{n}|=|(x_{n}-L)+L|\leq |x_{n}-L|+|L|<1+|L|}$

${\displaystyle M=\max \left(|x_{1}|,\ |x_{2}|,\cdots ,\ |x_{n}|,\ 1+|L|\right)}$

${\displaystyle |x_{i}|\leq M}$

### 保序性

${\displaystyle \lim _{n\to \infty }x_{n}=a}$
${\displaystyle \lim _{n\to \infty }y_{n}=b}$

${\displaystyle \epsilon ={\frac {a-b}{2}}>0}$，則由前提假設，存在 ${\displaystyle n_{1},\,n_{2}\in \mathbb {N} }$ 使任何 ${\displaystyle i\in \mathbb {N} }$ 只要 ${\displaystyle i>\max\{n_{1},\,n_{2}\}}$ 就有

${\displaystyle |x_{i}-a|<{\frac {a-b}{2}}}$
${\displaystyle |y_{i}-b|<{\frac {a-b}{2}}}$

${\displaystyle y_{n}

${\displaystyle y_{n}<{\frac {a+b}{2}}

${\displaystyle \epsilon -a
${\displaystyle \epsilon -b
${\displaystyle x_{i}>y_{i}}$

${\displaystyle 0

## 四則運算定理

${\displaystyle \lim _{n\to \infty }x_{n}=a}$${\displaystyle \lim _{n\to \infty }y_{n}=b}$，則

1. ${\displaystyle \lim _{n\to \infty }\left({{x_{n}}\pm {y_{n}}}\right)=a\pm b}$
2. ${\displaystyle \lim _{n\to \infty }{x_{n}}\cdot {y_{n}}=a\cdot b}$
3. ${\displaystyle b\neq 0,{y_{n}}\neq 0}$,則${\displaystyle \lim _{n\to \infty }{\frac {x_{n}}{y_{n}}}={\frac {a}{b}}}$.

## 参考文献列表

1. 华东师范大学数学系. 数学分析 第四版 上册. 北京: 高等教育出版社. 2010年7月第4版. ISBN 978-7-04-029566-5.