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?

Every sequence of integers $(a_k)_{k\in\mathbb{N}}$ can be represented as an infinite long string with the decimal representations of $a_k$ joined togather using "comma" as a separator.
For example, the Fibonacci sequence can be represented as a "string" beginning with characters "1,1,2,3,5,8,13,21,". 
If one view the resulting string as a sequence of bytes (e.g. encode the characters using ASCII encoding) and then view each byte as a sequence of $8$-bits, this representation will establish an injection from $\mathbb{N}^{\mathbb{N}}$ into $\{ 0,1 \}^{\mathbb{N}}$.
One the other hand,$\{ 0, 1 \}^{\mathbb{N}}$ can be viewed as a subset of
$\mathbb{N}^{\mathbb{N}}$. The natural embedding of $\{ 0, 1 \}^{\mathbb{N}}$
into $\mathbb{N}^{\mathbb{N}}$ is an injection in opposite direction.
By Cantor–Bernstein–Schroeder theorem, there is a bijection
between $\mathbb{N}^{\mathbb{N}}$ and $\{ 0, 1 \}^{\mathbb{N}}$
and hence $\displaystyle |\mathbb{N}^{\mathbb{N}}| = \left| \{ 0, 1 \}^{\mathbb{N}} \right|$.
