Application of Stone-Weierstrass to approximate $f\in C(X\times Y,\mathbb{R})$ where $X$ and $Y$ are compact Hausdorff spaces?

Question

On the wikipedia page for Stone-Weierstrass, the application section (http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Applications_2) says

If $X$ and $Y$ are two compact Hausdorff spaces and $f\colon X\times Y\to\mathbb{R}$ is a continuous function, then for every $\epsilon>0$ there exist $n>0$ and continuous functions $f_1,\dots,f_n$ on $X$ and continuous functions $g_1,\dots,g_n$ on $Y$ such that $\|f-\sum f_ig_i\|<\epsilon$.

Is there a way to tell that the subalgebra of finite sums of products of form $\sum f_ig_i$ separates points in $X\times Y$?

Cheers.

Answer 1

Hint 1: you don't even need the finite sum to separate a pair of points. Just a product of two functions. And one of them can be a constant.

Hint 2: recall Urysohn's lemma.