Point (a) of Exercise 5 Chapter 1 in Rudin's "Real and complex analysis" requires to prove the following:
If $f,g:X \to [-\infty, +\infty]$ are measurable then $\{x:f(x)<g(x)\}$ and $\{x:f(x)=g(x)\}$ are measurable.
Now the natural approach (and the one that you can find pretty much everywhere the exercise has been discussed) is to notice that $f-g:X \to [-\infty,+\infty]$ is measurable and then to write both sets as preimages of Borel subsets of $[-\infty, +\infty]$. However, Rudin has never proved that the sum of extended-real measurable functions is measurable (he has only proved it for non-negative extended-real functions). Now, I have found a proof of this fact by elementary means (see for instance here: https://www.cmi.ac.in/~prateek/measure_theory/2010-09-24.pdf), however that raises the question of why hasn't Rudin followed this more general road and instead only dealt, in a more obscure way, with a particular case in section 1.22. Is there something I am missing about his approach that makes it favourable?