How do I prove: If $A$ is an infinite set and $x$ is some element such that $x$ is not in $A$, then$ A\sim A\cup \left\{x\right\}$. How do I prove: If $A$ is an infinite set and $x$ is some element such that $x$ is not in $A$, then$ A\sim A\cup \left\{x\right\}$.

Hint: Since $A$ is infinite there is some $A_0\subseteq A$ such that $|A_0|=|\Bbb N|$. Show that $\Bbb N$ has the wanted property, conclude that $A_0$ has it, and then conclude that $A$ has it as well.
