Continuous - how can I combine these open sets

[CornMuffin]

continuous -- how can I combine these open sets

Homework Statement

let $X,Y$ be compact spaces
if $f \in C(X \times Y)$ and $\epsilon > 0$
then $\exists g_1,\dots , g_n \in C(X)$ and $h_1, \dots , h_n \in C(Y)$
such that $|f(x,y)- \Sigma _{k=1}^n g_k(x)h_k(y)| < \epsilon$ for all $(x,y) \in X \times Y$

Homework Equations

The Attempt at a Solution

$X,Y$ are compact which means that for all open covers of $X,Y$, there exists finite subcover.
So, i have been trying to think of a way to pick for all $x_0 \in X$ and $y_0 \in Y$, a function $g_{x_0} \in C(X)$ and $h_{y_0} \in C(Y)$ such that $f(x_0,y_0) = g_{x_0}(x_0)h_{y_0}(y_0)$ then there exists an open subset $U_{x_0,y_0}$ of $X \times Y$ such that $|f(x,y) - g_{x_0}(x)h_{y_0}(y)| < \epsilon$ for all $(x,y) \in U_{x_0,y_0}$. Then we can form an open cover of $X,Y$ and so there is a finite subcover, $U_1,\dots , U_n$

but i don't know how I can combine these open sets to get my functions $g_1,\dots ,g_n,h_1, \dots , h_n$ such that $|f(x,y)- \Sigma _{k=1}^n g_k(x)h_k(y)| < \epsilon$ for all $(x,y) \in X \times Y$

 

[Citan Uzuki]

Stone-Weierstrass theorem