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I was trying to come up with my own very simple proof of the Cauchy Criterion for Series $\left(\text{i.e., let}\, \{a_{n}\}\, \text{be a sequence in}\, \mathbb{R}. \text{The series}\,\sum_{n=0}^{\infty}a_{n}\, \text{is summable iff}\, \forall \epsilon > 0, \,\exists N \in \mathbb{N}\, \text{such that }\, \forall n > N,\, m \in \mathbb{N}, \, \text{we have}\, \displaystyle \left|\sum_{k=n}^{n+m} a_{k} \right| < \epsilon.\right)$.

The following is what I have done, and I was hoping if someone could tell me if it's correct, and if not, what I need to do to fix it:

The sum of the series $\displaystyle \sum_{k=0}^{\infty}a_{k}$ is by definition the limit of the partial sums $s_{k}$ (i.e., $\lim_{k \to \infty} s_{k})$. Recall from the Cauchy Convergence Criterion for Real Sequences that a sequence of real numbers $\{a_{n} \}$ converges iff it is Cauchy; i.e., iff $\forall \epsilon > 0, \, \exists N \in \mathbb{N}$ for which if $n$, $m \geq N$, then $|a_{m}-a_{n}|<\epsilon$.

So, the sequence of partial sums $\{ s_{k}\}$ converges iff it is Cauchy.

Is that all there is to this proof? Do I need to explicitly suppose $\{s_{k}\}$ converge and then show $\{ s_{k}\}$ is Cauchy? And then suppose $\{ s_{k} \}$ is Cauchy, then show it must converge? That seems a little redundant when citing this theorem; however, one thing I don't like is that I didn't show where the summation indices and limits come from here: $\displaystyle \left|\sum_{k=n}^{n+m} a_{k} \right| < \epsilon$ (namely, the $n+m$), and so I was wondering also if someone could help me with that.

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Your proof seems fine to me: The Cauchy criterion for series is the same as the Cauchy criterion for sequences (applied to the sequence of partial sums). Note that for $n \ge m$

$$|s_n-s_m|=\left|\sum_{k=m+1}^n a_k\right|$$

Now set $\tilde n=m+1$ and $\tilde m= n-m-1$, such that

$$\sum_{k=m+1}^n a_k = \sum_{k=\tilde n}^{\tilde n+\tilde m} a_k$$

See also https://math.stackexchange.com/a/1452800/32951

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