1. Let (S, d) be any noncompact metric space. Show that there exist bounded continuous functions fn on S such that fn (x ) ↓ 0 for all x ∈ S but fn do not converge to 0 uniformly. Hint: S is either not complete or not totally bounded.
2. Show that a metric space (S, d) is complete for every metric e metrizing its topology if and only if it is compact. Hint: Apply Theorem 2.3.1. Suppose d(xm, xn ) ≥ ε > 0 for all m /= n integers ≥ 1. For any integers j, k ≥ 1 let e jk (x, y) := d(x, x j ) +| j -1 - k-1|ε + d(y, xk ).
Let e(x, y) := min(d(x, y), inf j,ke jk (x, y)). To show that for any j, k, r , and s, and any x, y, z ∈ S, e js (x, z) ≤ e jk (x, y) + ers (y, z), consider the cases k = r and k /= r .