Reddit reviews The Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics)

Intro to Analysis - Covers roughly the first 7 chapters of Baby Rudin.

Measure Theory

Functional Analysis - Normed and Banach Spaces.

Point-Set Topology - Covers major topics from the general topology portion of Munkres.

Algebraic Topology - Covers Chapter 1 of Hatcher except for Van Kampen's Theorem.

Homology Theory - Covers Chapter 2 of Hatcher (starts with topics from Chapter 0 that will be needed).

Cohomology Theory - Chapter 3 of Hatcher.

Graduate Abstract Algebra (I, II) - Algebra by Lang.

Algebraic Number Theory (I, II, III, IV) - Lang's Algebraic Number Theory, and advanced topics.

Okay. The book ''Calculus'' by Michael Spivak link is an introduction to Calculus, but using precise/rigorous notions that do away with the vague and imprecise infinitesimals. This should be what you want.

If you want something a bit more hardcore, but self-contained, I can recommend that you look at Rudin's The Principles of Mathematical Analysis, which should be in any library.

Calculus done rigorously usually goes under the name of Analysis/ Real Analysis, there should be tons of books in any math library on this subject if you feel uncomfortable with the two above.

that's only for english language textbooks. in germany equivalent books are much cheaper. i dont think you will ever spend more than 80 dollars for the most expensive textbook.

entry level calculus will be more like 20-30 dollars, even new editions.

http://www.amazon.de/s/ref=nb_sb_noss_2?__mk_de_DE=%C3%85M%C3%85%C5%BD%C3%95%C3%91&url=search-alias%3Daps&field-keywords=analysis+1&rh=i%3Aaps%2Ck%3Aanalysis+1

http://www.amazon.de/Principles-Mathematical-Analysis-International-Mathematics/dp/0070856133/ref=sr_1_1?ie=UTF8&qid=1452638310&sr=8-1&keywords=rudin+principles

60 or 180 euros for rudin's book.

A much, much more inexpensive copy with the same content is also available.

Rudin is definitely the classic, but for a more contemporary and "friendlier" (but no less rigorous) introduction to real analysis, some people prefer the book by Pugh.

Edit: The two books cover pretty much the same material in the same order. I've heard Pugh described as "Rudin, with pictures"

In addition to upvoting, I wanted to reiterate the importance of two things 4Vesta said.

1. You MUST have the definitions down cold. If I wake you up in the middle of the night and ask you a definition, you must be able to write it down exactly right with no hesitation.
   
   
2. Math is not a spectator sport, you have to get in there and get your hands dirty by working tons of problems. An easy way to do that is by picking up analysis books cheap. I learned from Rosenlicht, which can be yours for less than $10. Rudin will cost you $15.