# 保西奴-華實斯定理

## 增減子數列存在定理

${\displaystyle x_{m_{1}}\geq x_{m_{2}}\geq \cdots \geq x_{m_{k}}\geq \cdots }$

${\displaystyle x_{m_{1}},{m_{2}},\cdots ,x_{m_{k}}}$

## 保西奴-華實斯定理

### 證明二

${\displaystyle a}$${\displaystyle b}$組成一個間距${\displaystyle I_{1}:=[a,b]}$

${\displaystyle n_{1}:=1}$，然後將${\displaystyle I_{1}}$斬開兩等分，即係${\displaystyle I_{1}'}$${\displaystyle I_{1}''}$。而將${\displaystyle (x_{n})}$大過一嘅項數${\displaystyle \{n\in \mathbb {N} :n>1\}}$分開做，

${\displaystyle A_{1}:=\{n\in \mathbb {N} :n>n_{1}=1,x_{n}\in I_{1}'\}\quad B_{1}:=\{n\in \mathbb {N} :n>n_{1}=1,x_{n}\in I_{1}''\}}$

${\displaystyle n_{2}}$定義為${\displaystyle A_{1}}$入面項數最細嗰一項。即係${\displaystyle A_{1}}$入面有${\displaystyle \{x_{3},x_{2},x_{5},x_{11},\cdots \}}$，咁${\displaystyle n_{2}:=2}$

${\displaystyle I_{2}:=I_{1}'}$，之後再將${\displaystyle I_{2}}$斬開兩等分，即係${\displaystyle I_{2}'}$${\displaystyle I_{2}''}$。而再將${\displaystyle A_{1}}$嘅數分開，

${\displaystyle A_{2}:=\{n\in \mathbb {N} :n>n_{2},x_{n}\in I_{2}'\}\quad B_{2}:=\{n\in \mathbb {N} :n>n_{2},x_{n}\in I_{2}''\}}$

${\displaystyle n_{3}}$定義為${\displaystyle A_{2}}$入面項數最細嗰一項。

${\displaystyle |x_{n_{k}}-\xi |\leq {\frac {b-a}{2^{k-1}}}}$