Question 5. Given a metric spaceX,p> (a) If the sequence (rn)EN CX is convergent, show that it is bounded. (b) If the sequence (n)EN CX is convergent, prove that it is Cauchy. Is the converse true? Justify your answer. (e) True or false? Justify your answer. If (zn)neN is a bounded sequence in X, then it has a convergent subsequence. (d) Given two sequences (n)neN. (Un)EN CX. Suppose that they converge to the same limit a X. Show that the metric distance p(x,y) → 0 as noo? Is it true that if p(zn. Un) → 0 as noo, then the two sequences have the same limit? Justify your answer.

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