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Question: do question 6 plz...

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5. Let (X,dx) and (Y.dy) be metric spaces. For (zw),(z,v) e X × Y, put d(z, y), (z, y)) = ldx (z,z)2 + dy(y, y) ]1 . (a) Check that d is a metric on XxY. (b) Check that for (Fm%), (z, y) E X x Y we have (zn,%) → (zw) in X × Y if and only ifFm → x in X and y, → y in Y.6. Suppose k E N, and (X1.4), . . . , (Xk, dk) are metric spaces. For x = (zı, . . . ,xkje X, x … × Xk Using induction or otherwise, generalize #5(a) and (b) to XX … x.Xk : Prove that d is a metric on X. lfr,Fn X, write x = (zi, . . .,Tk) and xn = (zni, . . . ,xnk) with li,TniE Xi for i = 1, . . . , k, and prove that n- in X if and only if ni as n for everyi E,.. ,k)

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