Banach Contraction Principle

So today we are going to look into Banach’s Contraction Principle(BCP) and Bessaga’s converse to BCP.

Theorem 1 Let \(X \not \emptyset\) be a metric space with metric \(d\) and let \(f : X \to X\) be a contraction map, i.e \(\exists \ \alpha \in [0,1)\) such that \(d(f(x),f(y)) \leq \alpha d(x,y) \ \forall \ x,y \in X\) Then \(f\) has an unique fixed point in \(\xi \in X\), moreover every iterate \(f^n\) of \(f\) also has the same fixed point \(\xi\).