Let \(\mathscr{F}:=\{F \subset \mathbb{N}: \# F<\infty\}\). Show that \(\# \mathscr{F}=\# \mathbb{N}\).

[ embed \(\mathscr{F}\) into \(\bigcup_{k \in \mathbb{N}} \mathbb{N}^{k}\) or show that \(F \mapsto \sum_{j \in F} 2^{j}\) is a bijection between \(\mathscr{F}\) and \(\mathbb{N}\).]