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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 $\ge p$.

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    Hi Marcus, welcome to Math.SE. This looks like a homework question; for such questions we expect you to follow [these guidelines from the FAQ](http://meta.math.stackexchange.com/questions/1803/how-to-ask-a-homework-question). Please edit your question accordingly.2012-09-19

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