Let $M = (X, d)$ be a D1107: Metric space.
A D62: Sequence $x : \mathbb{N} \to X$ is a Cauchy sequence with respect to $M$ if and only if \begin{equation} \forall \, \varepsilon > 0 : \exists \, N \in \mathbb{N} \, (n, m \geq N \quad \implies \quad d(x_n, x_m) < \varepsilon) \end{equation}

Let $M = (X, d)$ be a D1107: Metric space.
A D62: Sequence $x : \mathbb{N} \to X$ is a Cauchy sequence in $M$ if and only if \begin{equation} \forall \, \varepsilon > 0 : \exists \, N \in \mathbb{N} \, (n, m \geq N \quad \implies \quad x_m \in B_d(x_n, \varepsilon)) \end{equation}