In Descriptive Set Theory we often see the notion of encoding a real as a sequence of integers or natural numbers -- i.e. there obviously is a bijection according to ZF axioms. But how does it look like concretely? Anybody has seen a simple construction?

My own approach is by chain-fractions:

Let $q\in\mathbb{R}$ be the given real and now define the sequence $(z_i,q_i)$ by $$z_{i+1}=\begin{cases}[q_i]&\text{if } \{q_i\}\leq\frac{1}{2}\\ [q_{i}]+1&\text{else} \end{cases}$$ $$q_{i+1}=(q_i-z_{i+1})^{-1}$$ where $[q]$ is the next lower integer and $\{q\}=q-[q]$. (Hence $(q_i-z_{i+1})\in(-\frac12,\frac12]$ thereby absolute value of $q_{i+1}$, its reciprocal, is bigger than 2.) Now my bijection is mapping $q$ to the sequence: $$m_i=\begin{cases}z_i-2&z_i>0, i>1\\ z_i&i=1\\ z_i+2&z_i<0, i>1\end{cases}$$ with $i$ starting at 1 and above $q_0$ becomes the initial $q$. And the inverse of my bijection just calculates the chain-fraction:$q_{i-1}\in(z_i-\frac12,z_i+\frac12]$ with $q_{i-1}=z_i+q_i^{-1}$ step-wise narrowing down the real by a sequence of intervals each containing the next.

is there a paper or book covering my example? any other simple constructions?

are$\Bbb N^{\Bbb N}$! $\endgroup$