Construct a bijection given two injections

Question

I know that if there exists an injection $f: A \to B$, then $|A| \leq |B|$. Then trivially, $f: A \to B$ and $g: B \to A$ both being injections imply that $|A| = |B|$. However, the definition of $|A| = |B|$ means that there exists a bijection between $A$ and $B$. Is there any way to construct such a bijection from $f$ and $g$ or at least prove a bijection exists without using cardinality?

Answer 1

Yes, this is a known and very important result. With infinite sets, it is common that it is easy to find a pair of injections as you describe but not so easy to find an explicit bijection.

Schröder–Bernstein theorem