TITLE: Real analysis for a non-mathematician.
QUESTION [17 upvotes]: 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.

REPLY [2 votes]: I'm surprised no-one has mentioned Pugh's Real Mathematical Analysis. He has a whole chapter on formalism in math, specifically in analysis, where he guides you through thinking in inequalities and limits, and also is known for emphasizing pictures and geometric intuition, which is crucial for internalizing any proof. Highly recommend it -- very palatable for someone new to rigorous mathematics.