Cauchy criterion for series proof

Question

The series $\sum_{k=0}^{\infty} a_k$ converges if and only if $\forall\ \varepsilon > 0, ∃\ N$ such that if $m > n > N$ then $|\sum_{k=n+1}^{m} a_k |<\varepsilon.$

Could you please help me with that, how can I prove it for both sides?

Answer 1

Hint : Use the Cauchy criterion known for the sequences to the partial sums of your series.

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Let $S_n=\sum_{i=0}^n a_i$, with Cauchy criterion for the sequence $S_n$ you have :

$$S_n \text{ converges} \\ \iff \forall \epsilon>0,\exists N,\forall n\geq m>N,|S_n-S_m|<\epsilon \\\iff \forall \epsilon>0,\exists N,\forall n\geq m>N,\left|\sum_{i=m+1}^n a_i\right|<\epsilon$$