Let $A$ be countable. The set $[A]^n:=\{S \subseteq A|| S \mid=n\}$ is countable for all $n \in N, n \neq 0$

Question

I am solving the following problem:

Let $A$ be countable. The set $[A]^n:=\{S \subseteq A|| S \mid=n\}$ is countable for all $n \in N, n \neq 0$

I define the function $f:[A]^n\rightarrow N^n$ defined by $f(\left\{a_1, a_2, \ldots, a_n\right\}) = \left(a_1, a_2, \ldots, a_n\right)$, this function is 1-1 and $N^n$ is equipotent with $N$ then I have to $|[A]^n|\leq|N|$

how can I get $|N|\leq|[A]^n$|?

if I have $|N|\leq|[A]^n|$ then I could apply the Cantor-Bernstein theorem and $|N|=|[A]^n|$