6. Let (X1, d1) and (X2, d2) be metric spaces. Let X = X1 x X2 with the product topology and the metric d((x1, x2), (y1, yz)) = di (x1, y1) + d2(x2, 42)). a) If (X1, di) and (X2, d2) are complete, prove that (X, d) is complete. b) If (X1, di) and (X2, d2) are totally bounded, prove that (X, d) is totally bounded. c) If X1 and X2 are compact use a) and b) to prove that X is compact (this is a special case of Tychonoff's theorem, but with a nicer proof)