Let $(X,d)$ be a metric space. This space is compact if any sequence $x_n \subset X$ has a convergent subsequence.
This is how I'm given the definition of a compact metric space and it confuses me. How come a definition is not an "if and only if" statement and instead an "if" statement. This seems more of a theorem to me and seems like there could be some metric spaces which are compact but do not have a convergent subsequence.
Also, if say, $E \subset X$ is compact,then would the values of those convergent subsequences be a member of $E$?