Proof: If $K\subset \mathbb{R}$ is compact, then $\sup K$ and $\inf K$ both are in the set.

Question

1. If $K$ is compact then it is closed and bounded.
2. $\sup K=s$ and $\inf K = t$ are limit points of $K$, as it is possible to construct a sequence in $K$ that converges to them.
3. A closed set contains all it's limits points, hence $s,t$ is in $K$.

Am I missing something?

Answer 1

Your working seems fine.

You might like to explicitly state that $s$ and $t$ are finite because $K$ is bounded.