Q:

How to prove that the set of all not-empty and finite subsets of $\mathbb{N}$ is countable?

The set of all subsets of $\mathbb N$ is $ \mathcal{P}{(\mathbb N)} $ right?
So how can I prove that $ \mathcal{P}{(\mathbb N)} $ is countable?

A:

Let $A$ be a finite subset of $\mathbb N$. Write
$$ A = \{ a_1, a_2, \dots , a_n\}. $$
Let $B$ denote the set of all such finite subsets and define $f : B \rightarrow \mathbb N$ according to :
$$ \text{for } A = \{a_1, a_2, \dots , a_n\} : f(A) = 2^{a_1} + 2^{a_2} + \dots + 2^{a_n}$$
Then it is easy to see that $f$ is an injection of $B$ into $\mathbb N$, so $B$ must be countable.

