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 $\forall j, k > n$,

$$|a_j - a_k| < \epsilon$$