TITLE: Proof that there is a bijection, if there are injective maps in both directions
QUESTION [6 upvotes]: Let $A$ and $B$ be two sets.
Let $f:A\to B$ be injective such that $Im(f) \subsetneq B$.
Let $g:B\to A$ be injective such that $Im(g) \subsetneq A$.
Obviously $A$ and $B$ are not finite sets. Can we guarantee an existence of a bijection between $A$ and $B$?

REPLY [5 votes]: Yes. In fact, one can show that if there is an injective function from $A$ to $B$ and   an injective function from $B$ to $A$, then there is a bijection from $A$ to $B$. This is known as the Cantor–Bernstein–Schroeder theorem. (Of note is that its proof does not depend on the Axiom of Choice.)