Denumerable partition of a denumerable set where each set in the partition is denumerable.

Question

Suppose that a set $A$ is denumerable. Prove that there is a partition $P$ of $A$ where $P$ is denumerable and every $X \in P$ is also denumerable.

I can see that this can be done but I cannot figure out how to construct it where every $X \in P$ is denumerable and not finite. Any ideas here?

Answer 1

HINT: Let $A=\{x_n:n\in\Bbb N\}$, and use the inverse of the pairing function.