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 $\mathfrak{c}$.

$w\mapsto x+ \dfrac{y-x}{1+2^w}$ is an injective mapping in one direction.
(As $w\to\infty$, this function goes to $x$; as $w\to-\infty$, this function goes to $y$; for other values of $w$, it's between $x$ and $y$.)
$v\mapsto v$ is an injective mapping in the other direction.
