Post your favorite real analysis problem

[docnet]

TL;DR

I'm looking for good real analysis problems to hone my math skills. I didn't take a proper real analysis course during my BA, and skipped ahead to complex analysis and topology. And I felt that I didn't learn that much because my real analysis knowledge was shaky. And I figure it's time to catch up on what I'm lacking.

I figure one can dig around for plenty of quality problems online, but I'm hoping to skip right to the good stuff by you posting your favorite real analysis problems from college, internet, textbook, dream, imagination, etc. It would be so helpful to get some comments this time, and please don't post the solutions, I'm just looking for something to solve. Thanks.

 

[pasmith]

Cambridge past exam papers: https://www.maths.cam.ac.uk/undergrad/pastpapers/past-ia-ib-and-ii-examination-papers. You are looking for Analysis I in Part IA.

Cambridge Analysis I examples sheets: http://www.dpmms.cam.ac.uk/study/IA/AnalysisI/

 

[martinbn]

What do you mean by real analysis? Measures and integrals, or do you mean not complex?

 

[docnet]

What do you mean by real analysis? Measures and integrals, or do you mean not complex?

not complex, though I suppose there's no reason to exclude complex.

 

[FactChecker]

I don't think that real analysis is an essential prerequisite for complex analysis other than what you would know from a good calculus class. Real analysis concentrates on measure theory and sets/functions with bizarre properties, whereas complex analysis concentrates on functions with amazingly smooth behavior (EDIT: except around essential singularities).

(EDIT: On the other hand, I think that Real Analysis, measure theory, and Lebesgue integration is essential for any mathematician.)
I have always been a fan of Schaum's Outlines. They have a lot of solved problems and exercises. For their Real Analysis, I can only find an online PDF that requires some sort of payment.

 

[pasmith]

I think "real analysis", as a precursor to topology or complex analysis, means "single variable calculus, but done rigorously with proofs".