Convergence in product metric spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces . On the product $X\times Y$, define the metric $d_\infty\bigl((x,y),(x',y')\bigr)=\max\{d_X(x,x'),\,d_Y(y,y')\}.$ (Any equivalent product metric, such as $d_1=d_X+d_Y$, yields the same notion of convergence .)

Proposition (coordinatewise convergence): A sequence $((x_n,y_n))$ in $X\times Y$ converges to $(x,y)$ (with respect to $d_\infty$) if and only if $x_n\to x \text{ in } X \quad\text{and}\quad y_n\to y \text{ in } Y.$ Likewise, $((x_n,y_n))$ is Cauchy in $X\times Y$ iff $(x_n)$ is Cauchy in $X$ and $(y_n)$ is Cauchy in $Y$.

This proposition justifies treating product convergence as “simultaneous convergence of components.”

Proof sketch: By definition, $d_\infty\bigl((x_n,y_n),(x,y)\bigr)\to 0 \quad\Longleftrightarrow\quad \max\{d_X(x_n,x),d_Y(y_n,y)\}\to 0,$ which holds iff both $d_X(x_n,x)\to 0$ and $d_Y(y_n,y)\to 0$. The Cauchy statement is identical with $(x,y)$ replaced by $(x_m,y_m)$.