Let $X$ be a compact topological space and $Y$ a Hausdorff space. Let $C \subseteq Y$ be closed in $Y$ and $U$ an open set in $X \times Y$ which contains $X \times C$. Prove there exists an open set $V \subseteq Y$ such that $X \times C \subseteq X \times V \subseteq U$.
Here's what I tried.
Let $b \in C$ fixed. For each $x \in X$ find open sets $U_{x}$ and $V_{x}$ in $X$, $Y$ respectively containing $x$ and $b$. Then the colecction $\{U_{x}: x \in X\}$ covers $X$ so by compactness of $X$ , we have $X \subseteq \bigcup_{i=1}^{n} U_{i}$. From here how to obtain the open set $V$ ?, if we take $V$ as the finite intersection of the V_{i} then this won't satisfy $X \times V \subseteq U$. Also I don't see where to use the hypothesis that $C$ is closed.
I don't think the above approach works. Can you please help? Thanks.