Please provide the proof of this following question.

46.11 (Banach's theorem) Let f be a contraction mapping on a complete metric space M. (See Exercise 44.7 for the definition of a contrac- tion mapping.) (a) Let x, EM. Let Xn+1 = f(x,) for n = 1, 2, .. . . Prove that {x,} is a Cauchy sequence in M. (b) Prove that if x = lim,-> X, where {x,} is the sequence defined in part (a), then f(x) = x. (c) Prove that there is only one point x e M such that f(x) = x. Banach's theorem pro- vides another example of a fixed point theorem. (See also Exercise 42.12.)