 # Advanced Math Problems

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Question 5 Throughout this question we use the Euclidean metric for both [0, 1] and R. (a) For n ∈ N, define a function fn : [0, 1] → R by fn(x) = Xn i=0 2 −i (1 + sin(2ix)). (i) Prove that sequence (fn) has a pointwise limit, f : [0, 1] → R. Hint: use the monotone convergence theorem. (ii) By showing that for each n ∈ N, the function fn is Lipschitz, prove that each function fn is continuous. (iii) Determine whether the pointwise limit of the sequence (fn) is continuous. (b) For N ∈ N, dene FN : C[0, 1] → R by FN (f) = 1 N PN i=1 f( i N ). Show that FN is (dmax, d)-continuous on C[0, 1], where d is the Euclidean metric for R. Question 6 Dene a distance function on the plane, d : R 2 × R 2 → R by d(x, y) = ( d (2)(x, 0) + d (2)(y, 0), if x 6= y 0, if x = y. (a) Prove that d is a metric on R 2 . (b) (i) Let (xn) be a sequence of points R 2 . Show that (xn) is a d-convergent sequence if, and only if, either (xn) is eventually constant or (d (2)(xn, 0))n∈N is a real null sequence (in which case the d-limit is 0). (Recall that d (2) denotes the Euclidean metric for the plane.) (ii) Find Cl(R2,d)(A) when A = {(x1, x2) ∈ R 2 : x1 > 0}.

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