# Consequence of the Cauchy criterion for series

The Cauchy Criterion of a series is as follow:

Theorem: Let $$a_n$$ be a real sequence. Then the infinite series $$\sum_{k=1}^{+\infty}{a_k}$$ converges if only if for every $$\varepsilon > 0$$ there is an $$N \in \mathbb{N}$$ such that $$m > n > N$$ implies $$\left| \sum_{k=n+1}^{m}{a_k} \right| < \varepsilon .$$

From this result, how to demonstrate the following corollary?

Corollary: Let $$a_n$$ be a real sequence. Then the infinite series $$\sum_{k=1}^{+\infty}{a_k}$$ converges if only if for every $$\varepsilon > 0$$ there is an $$N \in \mathbb{N}$$ such that $$n > N$$ implies $$\left| \sum_{k=n+1}^{+\infty}{a_k} \right| < \varepsilon .$$

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By Cauchy's Theorem, for every $$\varepsilon > 0$$ there is $$N \in \mathbb{N}$$ such that $$n > N$$ imply $$\left| \sum_{k = n + 1}^{n+p}{a_k} \right| < \varepsilon,$$ for all integer positive $$p$$. Let $$S_{p}^{(n)}$$ represent the sequence of partial sums of $$\sum_{k = n + 1}^{n+p}{a_k}$$, when $$n > N$$ is fixed. Note that by the Principle of Mathematical Induction that $$S_{p}^{(n)}$$ is true for all $$p \in \mathbb{N}$$. Thus, $$\left| \sum_{k = n + 1}^{+\infty}{a_k} \right| < \varepsilon.$$
Conversely, let $$m, n$$ be positive integers such that $$m > n > N$$. Thus,
$$\left| \sum_{k = n + 1}^{m}{a_k} \right| \leq \left| \sum_{k = n + 1}^{+\infty}{a_k} \right| < \varepsilon.$$
Hence, by the Cauchy Criterion, $$\sum{a_k}$$ converges.