It is this fundamental mathematics that I consider to be part of a transition course. Such a course should also include all the nitty gritty about functions, relations, and a few other things as well as how to write a proof. Unfortunately, all too many schools still don't insure students know this stuff before they let them take higher math courses. I was unable to get the hang of higher math until I figured out on my own that this was what I was missing.
Ideally, this should be taught in freshman calculus. Limits would have been so much easier to understand. But, since freshman calculus courses are mostly populated by future engineers and engineers have a propensity to only want to know how to calculate things, perhaps professors don't try anymore. At least, this was my attitude when I was in my engineering phase.
"Analysis with an Introduction to Proof" by Steven Lay is a great book to have whether you know this transition material or not. I studied undergraduate analysis out of Rudin's book and Lay's book was an essential companion.