Banach Contraction Principle

Let (X,d) be a complete metric space, then each contraction map f : X → X has a unique fixed point.

Let h be a contraction constant of the mapping f . We will explicitly construct a sequence converging to the fixed point. Let x0 be an arbitrary but fixed element in X. Define a sequence of iterates {xn} in X by

xn = f(xn−1) (= f n(x0)), for all n ≥ 1.

Since f is a contraction, we have

d(xn, xn+1) = d(f(xn−1), f(xn)) ≤ hd(xn−1, xn), for any n ≥ 1.

Thus, we obtain

d(xn, xn+1) ≤ hnd(x0, x1), for all n ≥ 1.

Hence, for any m > n, we have

d(xn, xm) ≤  hn +hn+1 +···+hm−1 d(x0, x1) ≤ hn 1−h d(x0, x1).

We deduce that {xn} is Cauchy sequence in a complete space X. Let xn → p ∈ X. Now using the 

continuity of the map f , we get

p = lim n→∞ xn = lim n→∞ f(xn−1) = f(p).

Finally, to show f has at most one fixed point in X, let p and q be fixed points of f . Then,

d(p,q) = d(f(p), f(q)) ≤ hd(p,q).

Since h < 1, we must have p = q

The proof of the BCP yields the following useful information about the rate of convergence towards the fixed point.