# Definition of sequential compactness - Whats wrong with this alternation?

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 $A$.

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.