Exercise 8, § 4

Let $(X_i)_{1 \leq i \leq n}$ be a finite family of compact spaces, $X = \prod_{i=1}^{n} X_i$ their product, and for each index i, let A_i be a closed subset of X_i. Prove that every continuous numerical function in X which is zero in the union of the sets A_i × $\prod_{j \neq i} X_j$ (1 ≤ i ≤ n) can be uniformly approximated by sums of a finite number of functions of the form

where for each index i, u_i is a continuous numerical function in X_i and zero in A_i.