I am working on a problem I found that asks whether the set $S = \{x \in \mathbb{R} \mid x \ge 0\}$ is dense in $\mathbb{R}$.
The theorem I have been using states the following:
"$S$ is dense in $\mathbb{R} \iff \forall a,b \in \mathbb{R}$ with $a < b$, then $\exists x \in S$ such that $x \in (a,b)$."
Now my logic from what I have so far is basically that for any $a$ and $b$ you give me, I can find the midpoint between $a$ and $b$, which will consistently give me a positive real number. This feels like I'm just constructing sentences and not really "proving" it, however.
I am a bit confused and would like any words of advice.