Bolzano-Weierstrass Theorem/Lemma 2

Theorem
Let $S$ be a non-empty subset of the real numbers such that its infimum $\map \inf s$ exists.

Let $\map \inf s \notin S$.

Then $\map \inf s$ is a limit point of $S$.

Proof
The proof follows exactly the same lines as Lemma $1$.