Example: The sequence does not converge, but we can extract a convergent subsequence.

Therefore, using the Bolzano-Weierstrass theorem, there exists a convergent subsequence.

However, it is nearly impossible to actually list this subsequence. The Bolzano-Weierstrass theorem does guaranty the existence of that subsequence, but says nothing about how to obtain it.

The original sequence { sin(j) }, incidentally, does not converge. The proof of this is not so easy, but you might try to prove this yourself.