This question already has an answer here:
For every number $n\in\Bbb N$ let $\sigma_n$ be the binary sequence of $1$’s of length $n$ and then a $0$, this is – if you want to think of it that way – a representation of $n$ in base $1$. With a zero afterwards.
Now given a sequence $a=\langle a_n\mid n\in\Bbb N\rangle$ where $a_n$ is a natural number, we define the following binary sequence by induction.
$b$ is the sequence we have at the end of the induction process. Then $b\in 2^{\Bbb N}$, which is easy to see, as it is an infinite sequence of $0$-$1$ digits.
And I claim that the map sending $a$ to $b$ (where $a$ is a sequence of natural numbers, and $b$ is a sequence defined as above from $a$) is a bijection.
For this we just observe that the function decoding sequences of $1$’s between $0$’s is an inverse function. It’s a bit harder to write down formally, but I’ll try.
Let $b=\langle b_n\mid n\in\Bbb N\rangle$ be an infinite binary sequence. We define a sequence of integers.
I might be off with the indices in that second part, but that’s the general idea. We find out how to decompose $b$ into pieces looking like $\sigma_k$, then we take the $k$ from each piece.