Cardinality: $\left|\Bbb N^{\Bbb N}\right| = \left|\{0,1\}^{\Bbb N}\right|$ Let $F$ be the set of functions from $\Bbb N$ to $\Bbb N$ and $G$ be the set of functions from $\Bbb N$ to $\{0,1\}$. Prove that $|F| = |G|$.
What I tried doing is saying that every number in $\mathbb{N}$ maps to a bitstring, but I don't know how to construct from there?

\begin{align}
|\{0,1\}^{\mathbb{N}}|
&=2^{\aleph_0}\\
&\le\aleph_0^{\aleph_0}=|\mathbb{N}^{\mathbb{N}}|\\
&\le (2^{\aleph_0})^{\aleph_0}\\
&=2^{\aleph_0\aleph_0}\\
&=2^{\aleph_0}=|\{0,1\}^{\mathbb{N}}|
\end{align}
