How do I prove that an uncountable subset $S$ of $\mathbb{R}$ has the in-between property? The in-between property is that between any two distinct reals in the set, there is another real number. Also, $S$ has no discontinuities. It's not an interval such as $[0, 1] \cup [2, 3]$, for example. If $S$ were a closed, bounded interval, I can invoke the nested interval theorem to show the property holds. I'm not sure how to prove this for unbounded sets such as $\mathbb{R}^+$ or for bounded, open intervals. Thanks for your help.

Let $A$ be any subset of $\Bbb R$ with the property that if $a,b\in A$, and $a
Note that it is not true in general that if $S\subseteq\Bbb R$ is uncountable, then there is a non-trivial interval $I$ such that $S\cap I$ is densely ordered. The middle-thirds Cantor set $C$ is a counterexample: if $I$ is a non-trivial interval, and $I\cap C$ contains at least two points, then there are $a,b\in I\cap C$ such that $a