I've just acquired a copy of Principles of Mathematical Analysis by Ruden (3e) and, I must say, it is unlike any text I've ever read. Indeed, the level of abstraction and the amount of 'blanks' in the proofs have caused me to invest more time than usual into a book. Reading this has caused to me re-evaluate my command of mathematics. By this, I don't mean to convey the usual "I thought I was good, but realized I am not." Rather, I've found a whole new field for exploration, where I can become mathematically mature! Now, I have no delusions - I understand that this maturity only follows from dedicated study of the material. However, as I am self-taught, I don't really have any guidance in this matter, and do not wish to waste my time by practicing 'incorrectly,' or acting redundantly. It is my hope that the members of this forum can help me to develop an action plan, by answering the following questions: 1. What texts, at the same level of Ruden's book, could help cultivate mathematical maturity? 2. How exactly should I go about studying these books - in essence, how can I milk them for all they're worth? 3. Do you have any tips on writing proofs? Naturally, you will need to know my aspirations, and my current level of understanding. My main field of interest lies at the intersection of Differential Geometry and Differential Topology, though I also enjoy Projective Geometry and Algebraic Topology. My main aim is to have a profound, intuitive, and mathematically rigorous understanding of these topics (and, eventually, all of mathematics). Regarding my present ability, I understand all the mathematics necessary for graduate level physics, specifically in QFT and General Relativity. My strongest field is Riemannian geometry and Geometric Algebra, my weakest Real Analysis (for which I have only a basic exposure). Thank you in advance!