Let $(X,d)$ be a \reftext{def:metric-space-2026a}{metric space}, and let $(x_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence in $X$}. We say that $(x_m)$ is a Cauchy sequence in $(X,d)$ if for every real number $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that

for every $m,\ell\in\mathbb{N}$ with $m\ge N$ and $\ell\ge N$.