Let $g(x,y) \in C(X\times Y)$. prove for each $y\in Y$,we have $g(x,h+y) \to g(x,y)$ (when $h\to 0$) uniformly as $x$ range over compact set $K$.
I prove it as follows,and I was wondering whether I make a false proof:
Fixed $y$, w.l.o.g we may assume $|h|\le 1$,hence $g(x,y)$ is uniformly continuous over $K\times \overline{B_1(y)}$,hence for any $\epsilon>0$ if $d((x_1,y_1),(x_2,y_2))\le \delta(y)$ we have $|g(x_1,y_1)-g(x_2,y_2)|<\epsilon.$ for any $(x,y) \in K\times \overline{B_1(y)}$
Now come back to our uniformly convergence statement,for any fixed $y$,any $\epsilon>0$ taking $|h|\le \delta(y)$(the $\delta$ stated above),we have $|g(x,h+y) - g(x,y)|\le \epsilon$.
Is my proof correct?