Rudin's Principles of Mathematical Analysis has a really special place in my heart. Chapter 3 is great- it's a great reference for derivations of a lot of fundamental identities about limits used in undergrad calculus.

Chapter 4 is a great place to learn about topology for the first time.

In general, it kicks up the mathematical rigor you're used to a notch. Seeing ">" defined as "not <" really blew my mind when I first read it! "<" is just something that satisfies some axioms, like anything else in math.