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|f(x)-f(y)|<=|sinx-siny| implies id+f is monotone

Source: Romanian District Olympiad 2012, Grade X, Problem 1

October 9, 2018
functionalgebraPost 3 is the actual problem

Problem Statement

Let f:[0,∞)⟶R f:[0,\infty )\longrightarrow\mathbb{R} a bounded and periodic function with the property that |f(x)-f(y)|\le |\sin x-\sin y|, \forall x,y\in[0,\infty ) . Show that the function [0,∞)∋x↦x+f(x) [0,\infty ) \ni x\mapsto x+f(x) is monotone.
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