= Product metric
{title2=$d((x_1,x_2),(y_1,y_2))=d_1(x_1,y_1)+d_2(x_2,y_2)$}
For <metric spaces> $(X_1,d_1)$ and $(X_2,d_2)$, the sum metric on $X_1\times X_2$ is the displayed formula. Nonnegativity, symmetry and the <triangle inequality> follow by adding the respective inequalities in the two factors, and zero distance forces equality in both coordinates. It induces the finite <product topology>, as does the <maximum product metric>, since their distances satisfy $d_{\max}\leq d_{\mathrm{sum}}\leq2d_{\max}$. A sequence converges for this <metric> exactly when both coordinate sequences converge. Two successive extractions of a <convergent subsequence> therefore show that the product of two <sequentially compact spaces> is sequentially compact.
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