Content-Type: text/shitpost

[ This is another resurrection of a deleted Math Stack Exchange post. There's nothing really wrong with it, except that I feel like it's not of general interest. ]

Which parts of real-analysis are worth studying if you have already taken several calculus courses? I know that real-analysis is more 'rigurous', but still I wonder whether it is worth to again go over a lot of subjects that I already know from through calculus.

When I first entered university, shortly before classes began, I met with an professor whose task was to advise me on which classes to take in my first semester. After hearing me describe my background, which included passing the college-credit calculus exam at age fifteen, he suggested that I take real analysis.

“But I took that already,” I protested. “I had a two-semester course in real analysis at Columbia University last year. We used the little blue Rudin book. I got A’s.”

The professor said to me that analysis was a deep enough and rich enough subject that I would not be wasting my time to take it again, and that I would not be bored. I thought about this a little bit, and then I agreed that he was probably right. So I took the analysis course again. We used the same textbook, but I was not bored, and it was not a waste of time. It was an extremely good use of time; I have never regretted it.

So that's my answer about which topics of real analysis should be studied if you have already done calculus: all of them. You will not be bored, and it will not be a waste of time, because the answer is the same even if you have already taken real analysis.