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I've been trying to solve this question for a while now:

If $\{E_n\}$ is a sequence of totally bounded sets such that diam $E_n \rightarrow 0$, show that $\cup_{n=1}^{\infty} E_n$ is totally bounded.

I don't seem to understand what to do or how to solve this; please help.

Also, if the diam $E_n$ is going to 0, for any $r>0$, won't there occur a point after which the sets will be bounded by 1 ball of radius $r$, so if they all are disjoint, the total number of balls will be infinite, leading to a contradiction. I think my logic here is flawed somewhere.

Thank you so much

PCeltide
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    The statement isn't true. In $\Bbb R$ with the usual metric, taking $E_n={n}$ gives a counterexample. – David Mitra Jan 09 '20 at 16:46
  • That's a valid argument; so I believe the author must have made a mistake. Will my reasoning also have worked? – PCeltide Jan 09 '20 at 16:59
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    Just to say, I got this question from John B. Conway's book "A Course in Point Set Topology" pg. 29. – PCeltide Jan 09 '20 at 17:00

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