> 
> Let $f : [-1, 1] \rightarrow \mathbb{R}$ such that:
> 
> 
> 1. $|f(x) - f(y)| \le |x-y| \ \forall x,y$
> 2. $f(1)=f(-1)$
> 
> 
> Prove $|f(x) -f(y)| \le 1 \ \forall x,y$
> 
> 
> 

Making $y=0$ I get $|f(x) -f(0)| \le |x| \le 1$ but I cannot extend it to the required conclusion.