Just so that we can on the same page, I will present the couple definitions, let $X$ be the underlying metric space
'Bolzano-Weierstrass Property' is when every bounded sequence in $X$ has a converging subsequence.
Sequentially compact is when every sequence in $X$ has a converging subsequence.
Of course, sequentially compact is stronger than 'Bolzano-Weierstrass Property' but are there occasions where BWP will imply sequentially compact?
Best Answer
BWP implies sequential compactness iff $X$ is itself is bounded. This is because an unbounded sequence has a subsequence which has no convergent subsequence.
Suppose $(x_n)$is unbounded. Fix a point $x$. Then there is a subsequence $(x_{k_n})$ such that $d(x_{k_n},x) >n$ for all $n$. If this subsequence has subsequence converging to some $y$ we get $d(x,y)=\infty$, a contradiction.