Exercise If $X,Y $are two topological spaces and $X$ is not compact then, $Y$ is any space then $X \times Y$ is not compact.
My attempt: So we know that topological space is compact if every open cover has a finite subcover, so take an open cover of $X \times Y$. Say $U_\alpha$. We know that each open set in $X\times Y$ is of the form $U_i = V \times V'$, where V is open in X and V' is open in Y. Hence $U_\alpha = V_\alpha \times V'_\alpha$ ,where $V_\alpha $ and $V'_\alpha$ are open covers of X and Y respectively. But since X and Y are not compact we know that there does not exist finite subcover of neither $V_\alpha $ nor $V'_\alpha$. Hence we can not find finite subcover of $V_\alpha \times V'_\alpha$ and hence for $U_\alpha$.
Is it enough or it is not sufficient?