Please assist me with the following homework problem:
Let $X$ and $Y$ be metric spaces and suppose that $Y$ is compact. Let moreover $f: X \times Y \to R$ be a continuous function, and define a the function $F: X \to R$ by $F(x) = \sup \{ f(x,y) : y \in Y \}$. Prove that $F$ is continuous.
My thoughts so far:
(1) The compactness and the continuity of Y implies that $F$ attains a maximum value such that $\sup \{ f(x,y) : y \in Y \} \in R$ for each $x \in X$.
(2) The continuity of $f$ implies that $f(x,y)$ is bounded $\forall x \in X$ and $\forall y \in Y$.
$F$ is continuous if for every $\epsilon > 0$ there exists a $\delta > 0$ such that $|x-x'|< \epsilon$ whenever $d(x,x') < \delta$ $\forall x,x' \in X$.
To deduce the continuity of $F$ we just need to make sure that the domain of $F(x)$ exists for every $x \in X$ and that $F$ is bounded.