Diameter of Nested Sequence of Compact Set 
Possible Duplicate:
the diameter of nested compact sequence 

Let $(E_j)$ be a nested sequence of compact subsets of some metric space; $E_{j+1} \subseteq E_j$ for each $j$.
Let  $p > 0$, and suppose that each $E_j$ has diameter $\ge p$ . Prove that
$$E = \bigcap_{j=1}^{\infty} E_j$$
also has diameter $\ge p$.

For each $j$ pick two points $x_j, y_j \in E_j$ such that $d(x_j,y_j) \ge p$. Since $x_j \in E_1$ for all $j$, and $E_1$ is compact, the sequence $(x_j)$ has a convergent subsequence $(x_{\sigma(j)})$ say, and likewise $(y_{\sigma(j)})$ has a convergent subsequence $(y_{\tau \sigma(j)})$.
What can you say about the limits of these subsequences?
