Schröder-Bernstein Theorem

Let , be sets.
Assume that there are injections and .
Then there’s a bijection

Proof

We seek partitions of and of
s.t. and
Then

defines a bijection

For such partitions
Let’s define by
is a Complete Poset and is order-preserving.
By Knaster-Tarski fixed point theorem
there exists such that .
Then setting and and
gives the required partition.