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 $\ge p$.