Construct an injective function $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$ for some $a,b,c,d\in\mathbb{R}$

Question

Let $a,b,c,d\in\mathbb{R}$ such that $a<b$ and $c<d$ be given.
Construct an injective function $f:[a,b]\times[c,d]\rightarrow\mathbb{R}$.
My intuition is to construct a function $g:[a,b]\times[c,d]\rightarrow[0,1]$ and then show $g$ is injective. Then since there exists a bijection between $[0,1]$ and $\mathbb{R}$, then the composite must be injective.
However, I am having some difficulties to design a map from 2 dimension to 1. I have even built an injection $[0,1]\rightarrowtail[a,b]\times[c,d]$ and try to prove that surjection exist. But I still don't know how.
Hence I need some hints or guides for this one. Please don't just give an answer.
Many thanks,
S.

Answer 1

First you can built a bijection between $[a,b]\times [c,d]$ and $[0,1]\times [0,1]$ thanks to the map $$(x,y) \to \left(\frac{x-a}{b-a},\frac{y-c}{d-c}\right).$$ Now it remains to find an injection of $[0,1]\times [0,1]$ into $[0,1]$. You can for example use the famous Cantor's bijection. If $x = 0,x_1 x_2 x_3...$ and $y = 0,y_1 y_2 y_3 ...$, then you will associate $z=0,x_1 y_1 x_2 y_2 x_3 y_3 ...$ with the usual agreement that $0,99999... =1$.