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Let $E_{i}$ be a nested compact subsequence s.t $\forall i E_i\geqslant r$ for $r>0$. How can we show that this implies that $\bigcap_{i=1}^\infty E_i$ also has parameter bigger than $r$?

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I think that you mean diameter, not parameter. –  Brian M. Scott Sep 19 '12 at 5:46

1 Answer 1

I’ll do most of it and leave the last bit to you.

For each $k\in\Bbb Z^+$ there are points $x_k,y_k\in E_k$ such that $d(x_k,y_k)\ge r$. Since the sets $E_k$ are nested, all of these points are in the compact set $E_1$, so the sequence $\langle x_k:k\in\Bbb Z^+\rangle$ has a convergent subsequence $\langle x_{n_k}:k\in\Bbb Z^+\rangle$. Similarly, the sequence $\langle y_{n_k}:k\in\Bbb Z^+\rangle$ has a convergent subsequence $\langle y_{n_{k_i}}:i\in\Bbb Z^+\rangle$. Of course the sequence $\langle x_{n_{k_i}}:i\in\Bbb Z^+\rangle$ also converges, since it’s a subsequence of the convergent sequence $\langle x_{n_k}:k\in\Bbb Z^+\rangle$. And the sequence $\langle E_{n_{k_i}}:i\in\Bbb Z^+\rangle$ is still a nested sequence of compact sets of diameter at least $r$. Nobody wants to deal with three layers of subscripts, so for $i\in\Bbb Z^+$ we’ll let $F_i=E_{n_{k_i}}$, $u_i=x_{n_{k_i}}$, and $v_i=v_{n_{k_i}}$. Then $\langle F_i:i\in\Bbb Z^+\rangle$ is a nested sequence of compact sets of diameter at least $r$, $u_i,v_i\in F_i$ for $i\in\Bbb Z^+$, and the sequences $\langle u_i:i\in\Bbb Z^+\rangle$ and $\langle v_i:i\in\Bbb Z^+\rangle$ are convergent sequences in the compact set $F_1$.

What I just did in excruciating detail is normally handled simply by saying that without loss of generality we may assume that the sequences $\langle x_k:k\in\Bbb Z^+\rangle$ and $\langle y_k:k\in\Bbb Z^+\rangle$ converge.

Let $u$ be the limit of the sequence $\langle u_i:i\in\Bbb Z^+\rangle$ and $v$ the limit of the sequence $\langle v_i:i\in\Bbb Z^+\rangle$. Since $F_1$ is compact, and therefore closed, $u,v\in F_1$. But for any $n\in\Bbb Z^+$, $u_k,v_k\in F_n$ for all $k\ge n$, so $u,v\in F_n$. (Why? Make sure that you understand this step.) Conclude that $u,v\in\bigcap_{n\in\Bbb Z^+}F_n$, and show that $d(u,v)\ge r$.

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I am not sure why $F_i$ have at least r diameter? isn't there a possibility that the diameter will shrink if the some points are taken out? –  Student Sep 19 '12 at 11:58
    
@Student: No points were taken out. Each of the sets $F_i$ is one of the original sets. $\langle F_i:i\in\Bbb Z^+\rangle$ is a subsequence of $\langle E_i:i\in\Bbb Z^+\rangle$; I removed some of the sets from the sequence, but I didn’t remove any points from the sets. –  Brian M. Scott Sep 19 '12 at 20:07

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