For a compact set K, does $K\supset f(K)\supset f^{2}(K)\supset...$ hold for function f? Let $K$ be a compact normed space  and $f:K\rightarrow K$ such that $$\|f(x)-f(y)\|<\|x-y\|\quad\quad\forall\,\, x, y\in K, x\neq y.$$ 
Is it true that $K\supset f(K)\supset f^{2}(K)\supset...$?
How to show it?

If this were true, then all points of the space would be fixed: every set of the form $\{x\}$ is compact.
(and then the map would be the identity, which is impossible...)
