real analysis - Bijection from $mathbb R$ to $mathbb {R^N}$

How does one create an explicit bijection from the reals to the set of all sequences of reals? I know how to make a bijection from $\mathbb R$ to $\mathbb {R \times R}$.

I have an idea but I am not sure if it will work. I will post it as my own answer because I don't want to anchor your answers and I want to see what other possible ways of doing this are.

Answer

The nicest trick in the book is to find a bijection between $\mathbb R$ and $\mathbb{N^N}$, in this case we are practically done. Why?

$$\mathbb{(N^N)^N\sim N^{N\times N}\sim N^N}$$

And the bijections above are easy to calculate (I will leave those to you, the first bijection is a simple Currying, and for the second you can use Cantor's pairing function).

So if we can find a nice bijection between the real numbers the infinite sequences of natural numbers we are about done. Now, we know that $\mathbb{N^N}$ can be identified with the real numbers, in fact continued fractions form a bijection between the irrationals and $\mathbb{N^N}$.

We first need to handle the rational numbers, but that much is not very difficult. Take an enumeration of the rationals (e.g. Calkin-Wilf tree) in $(0,1)$, suppose $q_i$ is the $i$-th rational in the enumeration; now we take a sequence of irrationals, e.g. $r_n = \frac1{\sqrt{n^2+1}}$, and we define the following function:

$$h(x)=\begin{cases} r_{2n} & \exists n: x=r_n\\ r_{2n+1} & \exists n: x=q_n \\ x &\text{otherwise}\end{cases}$$

Now we can finally describe a list of bijections which, when composed, give us a bijection between $\mathbb R$ and $\mathbb{R^N}$.

1. $\mathbb{R^N\to (0,1)^N}$ by any bijection of this sort.


2. $\mathbb{(0,1)^N\to \left((0,1)\setminus Q\right)^N}$ by the encoding given by $h$.


3. $\mathbb{\left((0,1)\setminus Q\right)^N\to \left(N^N\right)^N}$ by continued fractions.


4. $\mathbb{\left(N^N\right)^N\to N^{N\times N}}$ by Currying.


5. $\mathbb{N^{N\times N}\to N^N}$ by a pairing function.



6. $\mathbb{N^N\to (0,1)\setminus Q}$ by decoding the continued fractions.


7. $\mathbb{(0,1)\setminus Q\to (0,1)}$ by the decoding of $h$, i.e. $h^{-1}$.


8. $\mathbb{(0,1)\to R}$ by any bijection of this sort, e.g. the inverse of the bijection used for the first step.