# Bolzano-Weierstrass theorem holds in any NLS. Is the statement true?

Bolzano-Weierstrass theorem holds in any NLS. (Normed Linear Space). Is this statement True?

If it is true how to prove it.

I know how to prove it in $\mathbb R$. I proved it in $\mathbb R$ by using supremum infimum of the set. But in NLS supremum infimum concept do not exist.

Consider the space $l_1$ and the sequence $e_n$ (the $n$th unit vector). This is bounded but had no convergent subsequence.