Let $X$ be a metric space with a metric $d$, let $E\subset X$. We have a function $f:E \rightarrow \mathbb R$ satisfying for some $M>0$: $ |f(x)-f(y)|\leq M d(x,y) \quad \text{for } x,y \in E. $
I wish to show that a function $ F(x)=\sup_{y \in E} [f(y)-M d(x,y)] $ for $x \in X$, is finite (that is $F: X \rightarrow \mathbb R$).