## Saturday, August 02, 2008

### solution

This is the solution for problem #1 in my previous post.

Let $\epsilon >0$, since $\{p_n\}$ is a Cauchy sequence there exist an $N_1$ such that if $d(p_n,p_m)<\frac{\epsilon}{2}$ if $n,m>N_1.$ Similarly since a subsequence converges to $p$ we have that $d(p,p_{n_i})<\frac{\epsilon}{2}$ when $n_i>N_2$. Now let $N$ be the larger of $N_1$ and $N_2$.

Now $d(p,p_n)\le d(p,p_{n_i})+d(p_{n_i},p_n)<\epsilon$.

Here is the solution for #2 in my previous post (this one took a little longer for me to think of).

Let $\{p_n\}$ be a sequence in $X$ such that $p_n\in E_n$. Now from properties of $\{E_n\}$ we have that $p_i\in E_j$ for all $j\le i.$ Now let $\epsilon>0$, and since $\displaystyle\lim_{n\to\infty}\text{diam}(E_n)=0$ we have that $d(p_n,p_m)<\epsilon$ for $n,m$ large enough. So $p_n\to p\in X$ since $X$ is complete. Additionally, $p$ is a limit point of each $E_n$, so $p\in E_n \forall n$.