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Let (X,d) be a complete metric space, and T a d-contraction on X, i.e., T: X...

Let (X,d) be a complete metric space, and T a d-contraction on X, i.e., T: X → X and there exists a q∈ (0,1) such that for all x,y ∈ X, we have d(T(x),T(y)) ≤ q∙d(x,y). Let a ∈ X, and define a sequence (xn)n∈Nin X by

x1 := a     and     ∀n ∈ N:     xn+1 := T(xn).

Prove, for all n ∈ N, that d(xn,xn+1) ≤ qn-1∙d(x1,x2). (Use the Principle of Mathematical Induction.)

Prove that (xn)n∈N is a d-Cauchy sequence in X. (Use Part (1) and the partial-sum formula for a geometric progression.)

As (X,d) is complete, Part (2) tells us that (xn)n∈N has a d-limit, which we shall denote by x*. Prove that x* is a fixed point of T, i.e., T(x*) = x*. (Use the Triangle Inequality.)

Prove that x* is the only fixed point of T, i.e., for all y ∈ X, if T(y) = y also, then y = x*.

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