Product of two compact spaces is compact

Yes, there is a mistake in this proof. The problem is that when you have your finite covers of $X$ and $Y$, it is true that the product of those open sets cover $X \times Y$ ; what is not true is that those products were in your original open cover!

Hope that helps,

You find that the product of a finite number of those projected open sets covers your space, but what you have found is not a subcover.