Series

为了应付接下来柯西序列的定义, 仅仅简单地定义一下序列. 更详细的定义, 见E.Ok: Series

Def 1: 有理数序列$(a_{n})_{n=m}^{\infty}:=f=\{(n,a_{n})\in I\times Q:nfa_{n}\}$. for index set $I=\{n\in Z:n\ge m \}$.

Def 2 (稳定): 如果$\forall a_{i},a_{j}\in \{a_{n}\}(d(a_{i},a_{j})<\epsilon)$, 则称该序列是$\epsilon$稳定的.

• 该定义的要求过于严格, 因为对初始值有要求

Def 2$\to$Def 3 (完美稳定): for $\epsilon >0$, $\exists N\in N\bigg(\forall n>N\bigg(\forall k\in N\big(d(a_{n},a_{n+k})<\epsilon\big)\bigg)\bigg)$. 即N之后的序列都是$\epsilon$稳定的, 则称该序列完美稳定.

Def 3$\to$Def 4 (Cauchy Series): $\forall \epsilon >0,\exists N\in N\bigg(\forall n>N\bigg(\forall k\in N\big(d(a_{n},a_{n+k})<\epsilon\big)\bigg)\bigg)$. 即对任意小正数, 柯西数列都是完美稳定的. 我们还没定义到实数, 所以$\epsilon$暂定为正有理数.

proposition 1: $\{a_{n}\}:=\frac{1}{n}$ 是柯西序列.

• 证: for $\forall \epsilon >0$, suppose $N=[\frac{1}{\epsilon}]+1$, we have: $\forall n> N(n>[\frac{1}{\epsilon}]+1)$. For $[\frac{1}{\epsilon}]+1>\frac{1}{\epsilon}$ (proposition 17 in From Natural to Rational), we have $\frac{1}{n}<\epsilon$. Q.E.D
• 好经典的极限证明

Def 5 (有界序列): $M\in Q_{+}$, 定义有界序列为: $\left\{ \begin{array}{rcl} Finite: \forall a_{i}\in\{a_{n}\}(|a_{i}|\le M)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ Infinite:\forall i\ge 1(\forall a_{i}\in (a_{n})_{n=1}^{\infty}(|a_{i}|\le M))\\\end{array} \right.$.

• 一个序列是有界的, iff $\exists M\in Q(该序列以M为界)$.

lemma 1: 任何一个有限序列都是有界的.

• 证: suppose for any series only contain one element $a_{0}$, it is bounded on $a_{0}$ as $|a_{0}|\le |a_{0}|$.

  • suppose for some n, the n-length series are bounded on M. This means $\forall a_{i}\in \{a_{n}\}(|a_{i}|\le M)$.
  • for $n+1$ length series, we would make sure the $1\to n$ elements are bounded on M, thus just need to compare the $|a_{n+1}|$ and M. However, because the M doesn’t need to precisely the upper bound of absolute value, then $\forall a_{i}\in \{a_{n}\}\le M+|a_{n+1}|$.

  Q.E.D

lemma 2: 任意一个柯西序列是有界的.

• 证: $\epsilon'=1\to \exists N\in N\bigg(\forall j,k>N(d(a_{j},a_{k})\le1)\bigg)\to \forall a_{j}\in(a_{n})_{n=N}^{\infty}(|a_{j}|<|a_{N}|+1)\to M=|a_{N}|+1$. $\to \forall a_{i}\in \{a_{n}\}_{n=1}^{N-1}(|a_{i}|\le\sum_{i'=1}^{N}|a_{i'}|=M')\to \forall a_{p}\in (a_{n})_{n=1}^{\infty}(a_{p}\le M+M')$

  Q.E.D

Def 6 (序列完美接近): $(a_{n})_{n=1}^{\infty}$和$(b_{n})_{n=1}^{\infty}$是完美接近的, 当对于一些$\epsilon>0\bigg(\exists N\in N\big(\forall j\ge N\big(d(a_{j},b_{j})\le\epsilon\big)\big)\bigg)$.

Def 7 (序列等价): $(a_{n})_{n=1}^{\infty}$和$(b_{n})_{n=1}^{\infty}$是等价的, 当$\forall \epsilon>0\bigg(\exists N\in N\big(\forall j>N\big(d(a_{j},b_{j})\le\epsilon\big)\big)\bigg)$.

proposition 2: $a_{n}=1+10^{-n}$ 等价于 $b_{n}=1-10^{-n}$.

• 证: 这里要用一下放缩法, 因为还没定义对数, 所以不能直接取对数的模. As $10^{n}>n\to10^{-n}<\frac{1}{n}\to2\cdot 10^{-n}<\frac{2}{n}$.

  $\forall \epsilon >0\bigg(N=[\frac{2}{\epsilon}]+1\to(n>N\to d(a_{n},b_{n})=2\cdot 10^{-n}\le\frac{2}{n}\le \epsilon)\bigg)$.

  Q.E.D

Problem Set For Series

5.2.1: 假设两序列等价, 如果其中一个序列是柯西序列, 当且仅当另外一个也一定是.

• 证: suppose $\{b_{n}\}$ is cauchy series, which means: $\forall \epsilon\exists N\forall n \forall k(|b_{n}-b_{n+k}|<\epsilon)\to \forall \frac{\epsilon}{4}\exists N\forall n \forall k(|b_{n}-b_{n+k}|<\frac{\epsilon}{4})$.

  $\to \forall \epsilon\exists N\forall n \forall k(|b_{n}-b_{n+k}|<\frac{\epsilon}{4})\to \forall \epsilon\exists N\forall n \forall k(|b_{n}-b_{n+k}|<\frac{\epsilon}{4})$. With the assumption that $(a_{n})\sim (b_{n})\to \forall \epsilon\exists N\forall n \forall k(|a_{n}-b_{n}|<\frac{\epsilon}{4})$. Thus we have: $\forall \epsilon\exists N\forall n \forall k (|b_{n+k}-a_{n}|<|b_{n}-b_{n+k}|+|b_{n}-a_{n}|<\frac{\epsilon}{2})$. Also, there is $\forall \epsilon\exists N\forall n \forall k(|b_{n+k}-a_{n+k}|<\frac{\epsilon}{2})\to\forall \epsilon\exists N\forall n \forall k(|a_{n}-a_{n+k}|<\epsilon)$. 简单来说就是对于任意小正数, 两序列永远比该正数还要接近.

  Q.E.D

5.2.2: 懒得写了