# ボルツァーノ-ワイヤストラスの定理

## Statement

すなわち $$\mathrm{(1)} \quad ( a_n )_{n\in \mathbb{N}},a_n\in\mathbb{R}$$ $$\mathrm{(2)} \quad \exists b,c\in\mathbb{R},\forall n\in\mathbb{N},b \lt a_n \lt c$$ $$\Rightarrow$$ $$\exists (n_i)_{i\in\mathbb{N}} \mathrm{s.t.} \quad n_i \in \mathbb{N}$$ $$\exists a \in \mathbb{R} \quad \mathrm{s.t.} \quad a = \lim_{i \rightarrow \infty} a_{n_i}$$

## Proof

(2) $\Rightarrow \exists b_0,c_0 \in \mathbb{R}$ s.t. $b_0 \lt (a_n) \lt c_0$
Define $(I_n)_{n \in \mathbb{N}}$ as $$I_0 = [b_0,c_0]$$ $$I_n = [b_n, c_n] = [b_{n-1}, \frac{b_{n-1} + c_{n-1}}{2}] \quad \mathrm{or} \quad [\frac{b_{n-1} + c_{n-1}}{2}, c_{n-1}]$$ where
$I_n$ has infinite elements of $(a_n)$
$\lim_{n \rightarrow \infty} |I_n| = \lim_{n \rightarrow \infty} (c_n - b_n) = \lim_{n \rightarrow \infty} \frac{1}{2^n}(c_0 - b_0) = 0$

$\exists a \in \mathbb{R}, \lim_{n \rightarrow \infty} b_n = \lim_{n \rightarrow \infty} c_n = a$
$\forall i \in \mathbb{N}, \exists n_i, a_{n_i} \in I_i$

$$\lim_{i \rightarrow \infty} a_{n_i} = a$$