Some problem about Cauchy sequence.

Question

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.

Answer 1

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

Answer 2

You may wish to look at Banach's Fixed Point Theorem, it's on Wikipedia.

Answer 3

The idea to show the existence of a fixed point of $f:X\to X$ is to pick an arbitrary point $x_0\in X$ and define a sequence $\{x_n\}_{n\in\mathbb{N}}$ in $X$ by the rule $x_n=f^n(x_0)$. Hint: $d(x_n,x_{n+1})\leq \gamma^n d(x_0,f(x_0))$. Can you now show existence of a fixed point of $f$ using (a)?

As for uniqueness, assume the existence of two distinct fixed points $x,y\in X$. What can you say about $d(x,y)$ using the given inequality?

Hope this helps!