Let $A$ and $C$ be infinite sets, with $C \subset A$, and suppose $|A|=|C|$. Now suppose there exists a set $B$ such that $C \subset B \subset A$.
Intuitively, $A$ and $B$ should have the same cardinality. I'm sure the proof is ridiculously easy but I haven't been able to find it. Can anyone help me, please?
Best Answer
Since nobody's answered this one yet, as Lord Shark suggests, the Schroeder-Bernstein theorem makes short work of this. Inclusion is an injective map $B\to A,$ whereas composing the inclusion $C\to B$ with the bijection $A\to C$ that exists by assumption gives an injective map $A\to B.$