I’m probably making this more complicated than it needs to be, but I’m trying to figure out why it is necessary to prove the convergence of a subsequence in proving the extreme value theorem (EVT). I typically see the EVT proved as follows:
Given an interval $f:[a,b]\rightarrow\mathbb{R}$:
- 1) prove that the sequence $f(x_n)$ converges to the supremum of the set $S=\{f:[a,b]\}$.
- 2) prove that a subseqeunce $f(x_{n_k})$ converges to $f(x_{0})$, and since a subsequence of a convergent sequence must converge to the same limit, then that means that $f(x_{0})=\sup S$.
Why is the second bullet point necessary, if we already proved that $f$ attains the supremum?