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The definition of sequential compactness:

A set $A$ in a metric space is sequentially compact if every sequence of elements of A has a convergent subsequence whose limit lies in $A$.

This alternation is wrong. Why?

A set $A$ in a metric space is sequentially compact if every converging sequence of elements of A has a limit that lies in $A$.

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The first definition is the right one to define sequential compactness of a set. The second definition is not an alternative one. It is the definition of a sequentially closed set $A$. The second pseudo-definition can not guarantee compactness. Take for instance $X= \mathbf R$ and let $A\subset X$ be an unbounded set.

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  • $\begingroup$ Thanks! On the other hand: Is every sequentially compact set sequentially closed? $\endgroup$ – stollenm Sep 24 '17 at 15:05
  • $\begingroup$ @stollenm A sequentially compact subset of a Hausdorff space is sequentially closed, as limits of convergent sequences are unique. $\endgroup$ – Henno Brandsma Sep 24 '17 at 21:57

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