I'm currently in an engineering program, so most of my mathematical education has been applied in nature (multivariable calculus, ODEs, PDEs, probability). The only real "theory"-based courses I've taken have been abstract algebra$^1$ and a proof-based differential equations class.
I'm looking to expand my mathematical horizons before graduate school, and I figured the two places that might be interesting for me (as well as useful!) are real analysis and differential geometry. I don't really have the space in my course schedule to take either of these, and the former is listed as a prerequisite for the latter at my university.
I know that Rudin's Principles of Mathematical Analysis is the crème de la crème for real analysis texts, but I've started reading a pdf of it and it's not only extremely dense (which I'm not quite used to), but I also don't think I have the mathematical maturity to grasp it.
I was wondering if anyone knows of any good texts where I can learn real analysis without presupposing a great deal of mathematical maturity. My foundations in calculus are quite strong, but my knowledge is that real analysis is only tangentially related. As a side question, I also want to ask if real analysis is really a prerequisite for differential geometry (I'm skeptical).
$^1$We covered what you would normally find in an undergraduate algebra class and also touched on a bit of Galois theory, but to be completely honest I wasn't all that comfortable with the few Galois theory lectures we had, anyway.
Best Answer
A rigorous calculus textbook would probably help you "get into" Rudin. An example is Lang. Spivak is also much liked, and there's Hardy's Course of Pure Mathematics whose first three editions are free and have lots of amusing exercises. However, Rudin has stuff that isn't in Lang or Hardy (or I assume Spivak) in chapters 9, 10, and 11.
As for whether real analysis is a prerequisite for differential geometry, I'm afraid that you need a mastery of the subject matter of Rudin chapter 9 (multidimensional derivatives, inverse functions, implicit functions) for modern differential geometry texts to make sense to you (e.g., Lee's Smooth Manifolds). An alternative might be a style of learning in which there is a separate course in 2- and 3-dimensional manifolds first; for this a common text is Manfredo do Carmo's Differential Geometry of Curves and Surfaces.
Independently of all the above, for math learning you need to (i) learn theorem-proof style and (ii) work exercises, writing down the proofs in "rigorous" mathematician style.