Answer to Question #133636 in Functional Analysis for PRATHIBHA ROSE C S

Question

Suppose that (xalpha)alpha element of j converges to x in X and (yalpha)alpha element of j converges to y in Y .show that (xalpha×yalpha) converges to x × y in X×Y

Answer 1

Let's suppose that $X$ is a set with element $x$ and that $Y$ is a set with element $y$. If we multiply two sets $X×Y$, all their elements will be in this new set. And if elements $x$ and $y$ converge in sets $X$ and $Y$ respectively, their product $x×y$ will converge in the set $X×Y$. Q.E.D.