Let $ f: M \to \mathbf{R}$ a $k$-Lipschitz function, i,e $ |f(x)-f(y)| \le k \cdot d(x,y) $ for every $x,y \in M$. Prove that $ \forall x\in M$ :
$ f\left( x \right) = \inf\limits_{y \in M} \{{f\left( y \right) + k \cdot d\left( {x,y} \right)}\} = \sup\limits_{y \in M} \{ {f\left( y \right) - k \cdot d\left( {x,y} \right)} \}. $ I have no idea how to prove this :/