$A$ is an infinite set.
Let $F$ be the set of all finite subsets of $A$, prove that $\operatorname{card}(F)=\operatorname{card}(A)$.
I want to prove this conclusion:
$\operatorname{card}\left(F\right)\leq \operatorname{card}\left(\bigsqcup_n \mathcal{F}\left(\left[n\right],A\right)\right)$
where $\left[n\right]$ is a n-tuple set and $\mathcal{F}\left(X,Y\right)$ is a mapping of $X$ onto $Y$.
Since after that $\text{LHS}\geq \operatorname{card}\left(A\right)$
$\text{RHS}\leq \operatorname{card}\left(\mathbb{N} \times A\right)\leq \operatorname{card}\left(A\times A\right)\leq \operatorname{card}\left(A\right)$.