A. Let (X, d) be a metric space. Define a diameter of a subset A of X and then the diameter of an open ball with center at xo and radios r >0 B. Use (A) to show that every convergence sequence in (X, d) is bounded

n= 3. Suppose that (X, d) is a metric space which is not compact. Let {an}_1 be a sequence of distinct points in X that has no convergent subsequences. For each n€ N, take ˜n > 0 such that Brn (an) Brm (am) = Ø) for n ‡ m. Let fn(2):= and rn - d(x, a) ™n + d(x, an) n f(x) = { fn(x), if a € Br. (an) for some n € N; otherwise. 9 Show that fn and f are continuous functions. Is f bounded?