Convergence in product metric spaces - Imported Knowlpedia Content

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 $Y$ .

Remarks

This proposition justifies treating product convergence as "simultaneous convergence of components."