(January 4, 2018 at 12:46 pm)Kernel Sohcahtoa Wrote:(January 4, 2018 at 8:25 am)polymath257 Wrote: Fair enough. But know that I am always willing to answer questions. If you don't want full solutions, I can usually point a direction.
Recently, I've become interested in real analysis and would like to learn material that would typically be covered in an intro to real analysis course (e.g., sequences, series, continuity, differentiability, integrability, and the real numbers). However, it seems that there's a lot of information to cover. With that said, in your opinion, what core concepts should I ensure that I understand in order to successfully complete the equivalent of an intro to real analysis course? Also, what books or resources would you recommend? Thanks.
P.S. Would you happen to have access to a course syllabus that provides a clear trajectory of how to proceed through the course along with the key topics and concepts to study (perhaps books to get and core exercises to work in those books)? Thanks again.
Off the top of my head: be conversant with 'epsilonics': doing epsilon-delta or epsilon-N proofs. Be able to use the definitions of limits and continuity to prove basic facts (sum of a limit is the limit of the sum, the product of two functions continuous at a point is continuous at that point, etc). Know about suprema/infima of subsets of R.
Know various aspects of compactness (in the real line, this is the same as being closed and bounded): Heine-Borel, convergence of Cauchy sequences, every bounded sequence has a convergent subsequence (but be aware that these do not carry over to general metric spaces). Know about uniform continuity and how it differs from continuity. Know, however, that a continuous function on a closed interval is automatically uniformly continuous.
Know and be able to prove results like the Intermediate Value Theorem and the Mean Value Theorem. Be able to use the MVT to derive standard Calc I results about increasing/decreasing functions. Be able to prove that a differentiable function on an interval such that f'(x)=0 everywhere must be a constant function.
Extra Credit: know that it is possible to be continuous everywhere, but differentiable nowhere. Be able to give explicit examples for lack of differentiability at a single point.
Know the basic procedure for defining the Riemann integral. Be able to show that any continuous function is integrable. Know how to prove the Fundamental Theorem of Calculus for continuous functions.
Often this first course in real analysis is used as an 'intro to proof' class as well. So a lot of time is spent learning how to negate statements with quantifiers (there exists, for every) and others (negate an implication, for example).
As for books. The old standard was Rudin's 'An Introduction to Real Analysis'. If you can find Stromberg's book 'An Introduction to Classical Real Analysis', it is possibly just a bit higher level than what you want, but it is an excellent source of exercises. It doesn't do the Riemann integral, however, but goes directly into the Lebesgue integral. I'll ponder a bit for others that might be good and I'll see if I can find a course outline for you.
