D 46. Given two metric spaces (M , d ) and (N . p ). we can dene a metric on the product M x N in a variety of ways. Our only requirement is that a sequence of pairs (an. x") in M x N should converge precisely when both coordinate sequences (0,.) and (x,.) converge (in (M , d) and (N , p ), respectively). Show that each of the following dene metrics on M x N that enjoy this property and that all three are equivalent: d] ((a. x). (b. y)) = do. b) + pu, y), d2((a. x). (b. y)) = (do. bf + per. mm, doo((a. X). (b. y)) = maXId(a. b). p(x. y)]- Henceforth. any implicit reference to \" e\" metric on M x N . sometimes called the product metric. will mean one of d. , (12, or doc. Any one of them will serve equally well; use whichever looks most convenient for the argument at hand