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Prove Cauchy's Convergence Criterion.

So I need to show that the partial sum $s_{n}=\sum^{n}_{i=0}a_{i}$ is Cauchy. $\rightarrow\forall\epsilon>0,\exists N$ s.t. $\forall n,m>N |s_m-s_n|<\epsilon$.

My guess is that I look at the fact that $s_n=\sum^{n} |s_n|=|a_{1}+...+a_{n}|$ and find a way to get $|s_m-s_n|<\epsilon$.

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    What do you mean by $s_n=\sum^{n} |s_n|=|a_{1}+...+a_{n}|$?2012-01-25

1 Answers 1

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Consider the partial sum in $\mathbb{R}$

$s_n = \sum_{i=1}^n a_i.$

Since $(\mathbb{R},|\cdot|)$ is complete, the partial sums converge iff they are Cauchy sequences. So for a convergent series this means that for any given $\epsilon > 0$

$\exists N_0 \in \mathbb{N} : \forall n,m > N_0 \implies \left|\sum_{i=0}^n a_i -\sum_{i=0}^m a_i\right| < \epsilon.$

Then all you have to do is observe that the difference for $n>m$ say is

$\sum_{i=0}^n a_i -\sum_{i=0}^m a_i = \sum_{i=m+1}^n a_i$