Let $M_1 = (X_1, d_1), \ldots, M_N = (X_N, d_N)$ each be a D1107: Metric space such that

(i)	$M = (X, d)$ is the D2509: Product metric space with respect to $M_1, \ldots, M_N$
(ii)	$x : \mathbb{N} \to X$ is a D62: Sequence in $X$

Then $x$ is a D65: Cauchy sequence in $M$ if and only if $\pi_1 x, \dots, \pi_N x$ are each a D65: Cauchy sequence in $M_1, \dots, M_N$, respectively.