Prove that the union of a finite or countable number of setseach of power $c$ (continuum power) is itself of power $c$.
To prove this I would start considering two sets $A$ and $B$ of power $c$. I would then try to show that $A$ and $A \cup B$ are equivalent using the Cantor-Bernstein theorem: given any two sets $M$ and $N$, suppose $M$ contains a subset $M_1$, equivalent to $N$, while $N$ contains a subset $N_1$, equivalent to $M$. Then $M$ and $N$ are equivalent.
Indeed, it is possible to establish a one-to-one correspondence $a \leftrightarrow a$ between each element of $A$ and of a subset of $A\cup B$, namely $A$. Now I would like to define a one-to-one relation from $A\cup B$ to $A$, but I don't know how to proceed.