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The product of two metric spaces (Y, dY ) and (Z, dZ) is the metric space...

The product of two metric spaces (Y, dY ) and (Z, dZ) is the metric space (Y × Z, dY ×Z), where dY ×Z is defined by dY ×Z((y, z),(y 0 , z0 )) = dY (y, y0 ) + dZ(z, z0 ).

Assume that (Y, dY ) and (Z, dZ) are compact. Prove that (Y × Z, dY × dZ) is compact.

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