Dear Physics Forum personnel, I am a college student with huge enthusiasm to the analysis and theoretical computer science. In order to start my journey to the real analysis. I am currently taking an introductory-analysis course (Rudin-PMA; I also use Shilov too) and linear algebra (Friedberg, but I mostly use Axler) courses, and my plan is to start studying for the real analysis and functional analysis right after the end of final exams. I have been looking for the good introductory books on the real analysis (as for the functional analysis, I bought Kreyszig), but there are just too many of them to select the few good ones....My future plan is to read Rudin-RCA during Summer of 2016, so my foundational plan is to use the Winter break and Spring Semester to read the introductory real-analysis book. I visited the mathematics library and went through a collection of books, and I did like Kolmogorov/Fomin, Carothers, and Stein/Sharkachi, but I am not sure if any of them is good for the pedagogical learning since I only read the few pages of the first chapters in those books. Could you inform me if any of them is good for the first introduction to the real analysis, and/or inform me some other books for the learning? Beside from Rudin, Shilov, and Axler, I read the topology sections of Simmons' "Introduction to Topology and Modern Analysis". Do I need additional background in the topology?