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 Post subject: Metric spacePosted: Tue, 20 Sep 2011 02:00:36 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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Location: Austin, TX
I found this cute, I got it from a friend on IRC:

Let be a compact metric space, and be distance preserving. Show is surjective.

Spoiler:
The way I know is by contradiction, but can you come up with another proof?

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 Post subject: Re: Metric spacePosted: Tue, 20 Sep 2011 05:17:23 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: NCTS/TPE, Taiwan
Spoiler:
There are at least two proofs by contradiction that I know of --- considering maximal cardinality of -separated set, or that , for fixed , having no convergent subsequence.

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 Post subject: Re: Metric spacePosted: Tue, 20 Sep 2011 05:20:21 UTC
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Spoiler:
From what I understand you can produce the sequence by filtering via the complements of your cover.

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