Spivak or Apostol to prepare for learning Baby Rudin on my own? I wish to learn real analysis on my own using the first 7 chapters from Baby Rudin. My goal is to have a good math background so as to study from PhD-level books in economics and finance on my own (e.g, stochastic calculus in finance, optimization, etc.). As an economics student, I find this kind of learning really challenging, so I appreciate it very much if you could give me some good suggestions/tips.
I have the following:
1) Apostol's single-variable calculus book without solutions manual
2) Spivak's single-variable calculus book + solutions manual
3) Time constraints and a rusty knowledlege of calculus (single-variable/multivariable calculus/LA)
If I have roughly 1-2 years ahead, I have to work full-time, and I have to choose only one of these 2 books, then which calculus book is better for self-study and for preparing for the first 7 chapters from Baby Rudin?
Thanks much
 A: First of all, the writing style of Baby Rudin is very concise and it's difficult to understand Rudin's Mathematical Analysis just by reading it one or two times without previous exposure to the topic. Therefore, I advise you to stay away from that book at all costs. There are better books available these days that have been written better and offer more insights and intuition about introductory Analysis such as Pugh's Mathematical Analysis.
One particular area that Rudin's Mathematical Analysis hasn't covered well is the equivalence of sequential compactness and compactness by open covers. It hasn't talked about the Lebesgue number of an open cover at all. But that's not all of it. You want to study financial mathematics, that means most importantly you need to learn measure theory which is covered in the last chapter of Baby Rudin and it is so abstract that when you read it for the first time, you will have absolutely no idea what he is talking about. Pugh's Mathematical Analysis is more verbose and it develops your intuition when it discusses measure theory.
Also, to understand financial mathematics, I assume you have to learn about stochastic integration and Ito's formula and maybe even Malliavin Calculus. This requires a fair amount of knowledge in "Real Analysis", not "Mathematical Analysis". There are topics that are not covered in a Mathematical Analysis course such as signed measures and Radon-Nikodym Theorem.
I believe Spivak's style of writing is more geometric than Apostol in general, therefore, I believe it is better if you want to cover the topics necessary for future understanding of differential manifolds. 
However, I advise you to choose Apostol's Calculus over Spivak and then continue to read Apostol's Mathematical Analysis. It is less difficult to read than Baby Rudin, it is verbose and covers a lot more details than Baby Rudin, it covers Bounded Variation functions that Baby Rudin does not cover, and it discusses Riemann-Stieltjes integral in such a detail that is way beyond someone's knowledge who has studied only Baby Rudin. It also discusses functional spaces in more details than Baby Rudin.
Therefore, my suggestion is like this: 1- Apostol's Calculus, Volume 1. 2- Apostol's Mathematical Analysis, excluding the chapters related to multi-variable analysis or Pugh's Mathematical Analysis excluding the chapter related to multi-variable calculus. 
Also, I advise you to review Calculus by solving like $10\%$-$20\%$ of the problems in Apostol's Calculus, volume 1. Having learned Calculus before, and exposure to its ideas and intuitions is definitely helpful, but do not spend too much of your time (like more than $2$ weeks) on Calculus because it's really not that necessary for learning Mathematical Analysis. 
A: Skip the calculus texts entirely. Analysis is a formalization of a lot of the concepts from calc; in Rudin's everything is built from a foundation so prerequisites are minimal (like high school pre-calc). 
I would buy Pugh's book on analysis as an alternative. It is always helpful to have a secondary source if something explained in the first isn't palatable. Good luck.
