Cauchy 序列的 Cauchy 序列
# Cauchy 序列的 Cauchy 序列
> $\{S_n\}$ 是收敛至$S$柯西列, 对每个$S_n$, $\{s_n^k\}$是收敛至$S_n$的柯西列, 请从$\{s_n^k\}$中挑出收敛至$S$的柯西列.
首先要做的是固定$n$, 均可挑出$\{s_n^k\}$的子列不妨仍记为$\{s_n^k\}$, 使得
$$
|s_n^k - S_n| \leq \frac{1}{2^K}, \forall k \geq K.
$$
挑取子列为$\{s_n^n\}$. $\forall \xi > 0$, 由$\{S_n\}$ 是收敛至$S$柯西列知$\exists N \in \mathbb{N}^+$, $\forall m, n > N_1$
$$
|S_m - S_n| < \frac{\xi}{2},
$$
$\exists N_2 \in \mathbb{N}^+$, $\forall n, \forall k > N_2$
$$
|s_n^k - S_n| \leq \frac{\xi}{8}.
$$
记$N = N_1 + N_2$, 则$\forall m, n > N$
$$
|s_m^m - s_n^n| \leq |s_m^m - S_m| + |S_m - S_n| + |S_n - s_n^n| \leq \xi.
$$
_关键是一致控制$\{s_n^k\}$是收敛至$S_n$的柯西列._
Sun, 18 Sep 2022 01:59:35 +0800
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