functions - Bijection from $(0,1]$ to $[0, infty)^2$

Define a bijection from $(0,1]$ to $[0, \infty)^2$

Route to follow,

A-) First define a bijection from $(0,1]$ to $(0,1]^2$

B-) Since there is a bijection from $(0,1]$ to $[0, \infty)$, namely $f(x) = (1/x) -1$, there is a bijection from $(0,1]^2$ to $[0, \infty)^2$

B says, if $f:A \rightarrow B$ is a bijection then there is a bijection $h:A^2 \rightarrow B^2$

Can anyone define me a function that satisfies A, and a function h for proof of B.

Rigor at elementary - intermediate analysis level will be appericiated.

Note: If possible I wonder the validity of infinite decimal approach for defining a function for part A.

Answer

For B): $h:=\langle a_1,a_2\rangle \mapsto \langle f(a_1),f(a_2)\rangle$, easy to see that it is a bijection, if $f$ was.

A) I think, can be valid, but should take care: first of all, I would apply $x\mapsto (1-x)$ because prefer $[0,1)$. Then, each $z\in [0,1)$ can be written in infinite decimal form, at most in two ways (since $0.1=0.99999\dots$), and choose the simpler one, so assume the digits are: $$z=0.a_1b_1a_2b_2a_3b_3\dots$$ Then $z\mapsto \langle 0.a_1a_2a_3\dots,\ 0.b_1b_2b_3\dots\rangle$ will almost be a bijection, but problems may occur only because of the above phenomenon, some pairs of numbers like ($0.0109090909\dots$ and $0.02$) will violate injectivity.

So, probably best is to consider first a bijection $[0,1)\to\{0..9\}^{\mathbb N}$, and then by this comb procedure conclude a bijection $\left(\{0..9\}^{\mathbb N}\right)^2 \to \{0..9\}^{\mathbb N}$ in an exact way.

The above map of decimal digits $g:[0,1)\to\{0..9\}^\mathbb N$ (which omits the sequences ending full of $9$'s from the range) is injective, and the number of omitted sequences is countable..

Note: Of course, we could also have used binary fractions and $\{0,1\}$ instead of $\{0..9\}$, but doesn't really matter.