What would you say are the prerequisites to the Analysis books? I did single-variable calculus and linear algebra in university a few years ago, but I have to admit that I'm a bit rusty.
Depending on the level of comfort or exposure you've previously had to proofs, it could be good to have a book covering introduction to proof handy. A book on counterexamples in analysis is also something I've seen recommended.
For proofs, something that covers direct proof, proof by contrapositive, proof by contradiction, and mathematical induction are good to be familiar with.
Delta-epsilon proofs are also good to have alternative or more accessible explanations to draw on.
If you're covering topics in multivariable calculus, then it might also be handy to cover some of the calculations from a calc 3/4 book to see implementations of it.
(I'm of the opinion that the usual Calc1+2+3 sequence should be scrapped in favor of everyone taking "Advanced Calculus" first. It should probably even be taught in 10th/11th grade in place of the dreadful "pre-calc" courses many schools have. Calculus didn't click for me until my first proof-based course. That's also when I learned that a "proof" is just a detailed explanation of exactly why something is true.)