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?

This is not true.  Take $f(x) = x^2$ on $[0,\frac{1}{2}]$.  Then $K = [0,\frac{1}{2}]$, and $f(K) = [0,\frac{1}{4}]$.
