Possible Duplicate:
Prove the map has a fixed point

Let $X$ be a compact metric space and $f:X\rightarrow X$ be a continuous function such that $d(f(x),f(y))<d(x,y)$ for every $x,y\in X$ such that $x\neq y$.

a) Prove that $f$ has a unique fixed point by minimizing $d(f(x),x)$. b)Prove that for any $x\in X$, the sequence $x,f(x),f(f(x)),\ldots, f^n(x),\ldots$ converges to the fixed point.