Question

What is an example of a sequence doesn't contain any convergent subsequences, but does have a subsequence that's bounded?

Answer 1

Nvm, it's impossible lol.

If a sequence is bounded, then it has a subsequence that converges, by the Bolzano Weissterthingymajiggy theorem, then that subsequence is also a subsequence of the original sequence, and that contradicts the fact that the original subsequence can't have any convergent subsequences. Easy peasy.