Thanks for providing another suggestion on the real analysis front. Really, I agree with you on the Rudin front. Honestly, I think that real analysis is the l

Thanks for providing another suggestion on the real analysis front. Really, I agree with you on the Rudin front. Honestly, I think that real analysis is the largest barrier for people trying to self study applied math. I'll mention here that I actually had a total of five semesters in real analysis all said in done. The first was an intro to real analysis course that used "Introduction to Real Analysis" by Bartle and Sherbert. The second was a more advanced real analysis course that was two semesters long using "Real Analysis and Foundations" by Krantz. When I entered my Ph.D, I was given an initial evaluation and the committee found my real analysis still too weak, which necessitated another two semesters using baby Rudin. Only at that point, did it really take and it's not been an issue since. Anyway, mostly I wanted to echo that I agree with you. I think Rudin is the metric to say good enough, but I was not able to get through it on first go.