fixed point of T: As a contraction mapping, T is continuous, so bringing the limit inside T was justified. Lastly, T cannot have more than one fixed point in (X,d), since any pair of distinct fixed points p1 and p2 would contradict the contraction of T: Applications A standard application is the proof of the Picard–Lindelöf theorem about the existence and uniqueness of solutions to certain ordinary differential equations. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which changes continuous functions into continuous functions. The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point. One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are bi-lipschitz homeomorphisms. Let Ω be an open set of a Banach space E; let I : Ω → E denote the identity (inclusion) map and let g : Ω → E be a Lipschitz map of constant k < 1. Then Ω′ := (I+g)(Ω) is an open subset of E: precisely, for any x in Ω such that B(x, r) ⊂ Ω one has B((I+g)(x), r(1−k)) ⊂ Ω′; I+g : Ω → Ω′ is a bi-lipschitz homeomorphism; precisely, (I+g)−1 is still of the form I + h : Ω → Ω′ with h a Lipschitz map of constant k/(1−k). A direct consequence of this result yields the proof of the inverse function theorem. It can be used to give sufficient conditions under which Newton's method of successive approximations is guaranteed to work, and similarly for Chebyshev's third order method. It can be used to prove existence and uniqueness of solutions to integral equations. It can be used to give a proof to the Nash embedding theorem. It can be used to prove existence and uniqueness of solutions to value iteration, policy iteration, and policy evaluation of reinforcement learning. It can be used to prove existence and uniqueness of an equilibrium in Cournot competition, and other dynamic economic models. Converses Several converses of the Banach contraction principle exist. The following is due to Czesław Bessaga, from 1959: Let f : X → X be a map of an abstract set such that each iterate fn has a unique fixed point. Let then there exists a complete metric on X such that f is contractive, and q is the contraction constant. Indeed, very weak