Help using the C-S-B theorem Let $\Bbb R$ denote the set of real numbers.  Let $H\subseteq\Bbb R$ and assume that there are real numbers $a,b$ with $a>b$ such that the open interval $(a,b)$ is a subset of $S$. Prove that the cardinality of $H$ equals $\mathfrak{c}$.

You can also use $\arctan:=\tan^{-1}$ for a map from $(-\pi/2, \pi/2) \rightarrow \mathbb R$ to inject, and , say, $f(x)=x$ in the other direction.
