continuous -- how can I combine these open sets

CornMuffin

1. The problem statement, all variables and given/known data
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 ##


2. Relevant equations




3. 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