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# Real Analysis

## Sequences

### Cauchy Sequences

A Cauchy sequence is a sequence in a metric space such that subsequent elements become arbitrarily close to one another. Formally, a sequence of real numbers $$(a_i)_{i \in \mathbb{N}}$$ is Cauchy if $$\forall \epsilon > 0, \exists n \in \mathbb{N}$$ such that $$\forall j, k > n$$,

$|a_j - a_k| < \epsilon$