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 \(\forall j, k > n\),

\[|a_j - a_k| < \epsilon\]