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[Kernel Sohcahtoa]

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.

[1] 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.

[2] 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.

[3] 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.

[4] 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.

[5] 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.

[6]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.

Thanks for your thoughtful reply, polymath.   I've actually spent a great deal of time honing my understanding of proofs: I have worked numerous exercises in Solow's How to Read and Do Proofs, Hammack's Book of Proof, and Chartrand's Mathematical Proofs, all of which cover the basic proof writing techniques.  Regarding part [6] of your post, I gained a good understanding of learning how to prove statements  containing quantifiers (the choose, construction, and specialization methods) and how to negate statements containing quantifiers (and implications), especially nested quantifiers, via Solow, which IMO, is very useful when doing a proof by contradiction, contrapositive, or elimination.  

Regarding part [1] of your post, I gained a good understanding of epsilon-delta proofs and epsilon-N proofs via Chartrand's section on calculus proofs and was able to understand the proofs of the basic limit properties: I really think it is cool yet challenging to construct that respective delta or N value (after arbitrarily choosing an epsilon value greater than zero) that will ultimately result in the function or sequence being within epsilon of the limit. In addition, via Chartrand,  I did a number of exercises pertaining to properties of real numbers, especially the triangle inequality.  Also, via Abbott's Understanding Analysis (I own this), I have gained an understanding of suprema and infima of subsets of R and was able to wrap my head around the nested interval property, Archimedean property, and axiom of completeness. 

It sounds like most of my time will be spent learning parts [2], [3], [4] and [5] of your post (since I learned calculus, I always wanted to learn why integrals work).  Also, I do own Rudin's Principles of Analysis; it seems quite advanced, but nevertheless, it does look like it is an invaluable resource
. I also own Introduction to Real Analysis by Schramm (I like his proof writing style, as it resembles Solow); Introduction to Analysis by Rosenlicht; Foundations of Analysis by Johnsonbaugh, and How to Think About Analysis by Alcock.  With that said, thanks for taking the time to respond to me. 

P.S. Since my goal is to learn material covered in an intro to real analysis course, would you recommend that I also study metric spaces or is this topic more advanced than the material that is usually covered in an intro level course? Thanks again for your time and attention.