So, I have a problem that I'm working on and I can't seem to figure it out. We just started product topologies, so many properties are still new or unknown to me. Here's the problem:

Let $\displaystyle (X,\Omega)$ and $\displaystyle (Y,\Theta)$ be topological spaces. If $\displaystyle A \subseteq Y$ is compact relative to $\displaystyle \Theta$ and $\displaystyle x \in X$, show that $\displaystyle \{x\}\times Y$ is compact relative to the product topology on $\displaystyle X\times Y$.

I'm not seeing why $\displaystyle A$ being compact is sufficient for the whole product to be compact... Any help would be appreciated.