8. Let X and Y be metric spaces, E <;:; X, and I : E -+ Y.

(a) If I is uniformly continuous on E and Xn E E is Cauchy in X, prove that I(xn ) is Cauchy in Y.

(b) Suppose that D is a dense subspace of X; i.e., D c X and D = X. If Y is complete and I : D -+ Y is uniformly continuous on D, prove that I has a continuous extension to X (e); i.e., prove that there is a continuous function