Some problem about Cauchy sequence. I cannot solve this:
Let X be a complete metric space with a metric d.
(a) Suppose that the sequence $x_{n}$ in X satisfies
$\sum_{n=0}^{\infty}d(x_{n},x_{n+1})<\infty$
Show that $x_{n}$ converges
(b) Suppose that X is nonempty and f is a function from X to itself such that
$d(f(x),f(y))\leq \gamma d(x,y),  x,y\in X $
for a consistant $0< \gamma < 1$. Prove that there is a point $p_{0}\in X$ such that $f(p_{0})=p_{0}$ and there is only one such point. (Hint: consider the sequece given by $x_{n+1}=f(x_{n})$ and show its convergence.)
I can solve part (a) by using $d(x_{n},x_{n+1})$ converges to 0.
But I cannot solve part (b), and escpecially I cannot express f. 

For (a), first show that the sequence is Cauchy (see here), then you can conclude. 
