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I have to show that if a metric space is path connected and countable then it is complete. I'm pretty lost where to start this at all. I have the basic definitions of complete, path-connected, compact and sequentially compact spaces.

Any help how to do this would be great (this is a past paper question-non assesed, just for practice so I think it should be reasonable simple)

Thanks very much for any help

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    Actually, your version is simpler. The image will necessarily be a singleton for a non-empty countable space, meaning $d(x,y)=0$ for all $y$ in the space, and so there is only one point in the space by metric properties.2012-05-04

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