The cardinality of the union of two infinite sets of equal cardinality 
Prove that if A is an infinite set and $|A| = |B|$ then $|A| = |A \cup B|$.

I don't know how to get an injection from $A∪B$ to $A$. I thought about considering $A\times B$, but don't see how that could work. 

The equality of $|A|$ and $|B|$, and Cantor-Schroeder-Bernstein theorem allows us to reduce to proving $|A|=|A\coprod A|$. This is accomplished by writing $A=\coprod_{S}\mathbb{N}$, and noting $A\coprod A=\coprod_{S}\mathbb{N}\coprod\coprod_{S}\mathbb{N}=\coprod_{S}(\mathbb{N}\coprod \mathbb{N})$, we are reduced to $A=\mathbb{N}$. There is a bijection $\mathbb{N}\to \mathbb{N}\coprod \mathbb{N}$ given by enumerating even and odd numbers. 
