Is every locally compact, complete, bounded space compact?
It seems that this fact is implicitly used in the definition of GromovHausdorff convergence for pointed spaces on wikipedia, but I'm not able to prove it.
Is every locally compact, complete, bounded space compact? It seems that this fact is implicitly used in the definition of GromovHausdorff convergence for pointed spaces on wikipedia, but I'm not able to prove it. 

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I don't believe this is true. For example, take an infinite set and put discrete metric on it, that is, $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ if $x\not=y$. Then I believe this is locally compact, complete and bounded, but it is not compact. Did I understand your question correctly? 


A metric space is compact if and only if it is complete and totally bounded. This is the HeineBorel theorem. Is this what you are looking for? 

