Writing Errands

Please help me prove the following about sequences of continuous functions
Let X, Y be compact metric spaces and consider the metric space X x Y equippedwith the metric d((x1, y1), (x2, y2)) = dx(x1, 72) + dy (y1, y2). Prove that for everyf EC(X xY) and every e > 0, there exist functions gi, . . . On E C(X) and hi, . .. hn EC(Y) such thatnIf (x, y) -gi (x )hi(y) <e for all x EX, yEY.j=1

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