Let $X $ be a compact metric space, and $Y $ be a metric space. Let $F : X × X \to Y $ be a continuous mapping. Let $f_y (x) = F\left(x,y\right) $. Show that $A = \{ f_y : y \in X \}$ is equicontinuous.
I know that a continuous function from a compact metric space is uniformly continuous. So given $ \epsilon \gt 0$, I can produce a $\delta_y \gt 0$ which doesn't depend on $x $ such that the definition of continuity holds for $f_y $. Now if I can somehow produce finite number of $\delta_{y_i}$'s, I will be done. But I cannot see any way of getting finite number of $y_i $'s. Any hint is appreciated. Thanks.
Hint: Use the uniform continuity of $F$ on $X\times X.$