Construct an injective function $f:[a,b]times[c,d]rightarrowmathbb{R}$ for some $a,b,c,dinmathbb{R}$

Let $a,b,c,dinmathbb{R}$ such that $a<b$ and $c<d$ be given.
Construct an injective function $f:[a,b]times[c,d]rightarrowmathbb{R}$.
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.