Question

A very simple math question involving sequences and convergence:

Answer 1

Let \(a_{n}\) be a bounded sequence, and define the set \(S = \left\{ x \in \mathbb{R} , x < a_{n} , \forall n \in \mathbb{N} \right\}\)
Show that there exists a subsequence \( a_{n_{k}} \) that converges to the least upper bound of \(S\)


My problem with this question is that I think it's false:
Consider this:

As you can see, this sequence is increasing starting from the lower-bound and going towards the upperbound. SUBSEQUENCES THEN MUST ALSO BE INCREASING....nvm