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I read that in a metric space compactness and sequential compactness mean the same thing. In $\Bbb R$ is sequential compactness equivalent to compactness? I see some definitions of Heine–Borel theorem use compactness, and others use sequential compactness.

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$\mathbb{R}$ and $\mathbb{R}^n$ are metric spaces (with metric given by $d(x,y)=\lvert x-y\rvert$).

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You probably mean $||x-y||$ in $\mathbb R^n$. –  user18119 Oct 10 '12 at 20:20
    
@QiL Yes, sort of, though it is not at all uncommon to use the $\lvert\cdot\rvert$ notation even in $\mathbb{R}^n$. It depends on context. Neither notation is clearly wrong. –  Harald Hanche-Olsen Oct 11 '12 at 8:08

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