Cauchy criterion for convergence

If a sequence converges, it is the same as saying it matches the Cauchy criterion for convergence.

Cauchy Sequence

A sequence

$$(a_n)^\infty_{n=1}$$ is Cauchy if:

$$\forall\epsilon>0\exists N\in\mathbb{N}:n> m> N\implies d(a_m,a_n)<\epsilon$$

Theorem

A sequence converges if and only if it is Cauchy

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TODO: proof, easy stuff

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[Expand]Real Analysis

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Functional Analysis

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