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Prove that, there exists a x in S such that f(x)=x

Source: MTRP 2018 Class 11-Short Answer Type Question: Problem 4 :-

February 17, 2021
functionalgebra

Problem Statement

Suppose SS is a finite subset of R\mathbb{R}. If f:S→Sf: S \rightarrow S is a function such that, ∣f(s1)−f(s2)∣≤12∣s1−s2∣,∀s1,s2∈S \left|f\left(s_{1}\right)-f\left(s_{2}\right)\right| \leq \frac{1}{2}\left|s_{1}-s_{2}\right|, \forall s_{1}, s_{2} \in S Prove that, there exists a x∈Sx \in S such that f(x)=xf(x)=x
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