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I am planning to self study Real Analysis over the summer by following Abbott's Understanding Analysis and another set of lecture notes on an undergraduate single-semester Real Analysis course. Before I start I want to ask which areas in a typical undergraduate course in Real Analysis do learners tend to have particular trouble grasping and which areas are required to be fundamentally understood. Answering this would help me know which areas I need to especially look out for while learning.

Thank you

linthevin
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  • My biggest difficulty was understanding the power of subsequences of convergent sequences and learning how to use those well. Especially in higher levels of real analysis where I have come across several questions where the answer requires taking subsequences of subsequences of a convergent sequence. It can get really confusing it you lose track of your indices. –  May 18 '20 at 20:40
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    In my experience, the toughest hurdle for students in a class like this is not any real analysis topic per se, but general familiarity with logic and proof techniques - sometimes called "mathematical maturity". – Jair Taylor May 18 '20 at 20:42
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    For some $\epsilon-\delta$ style proofs you have to keep a number of variables in your head and remember whether they are universally or existentially quantified, and whether they are given by hypothesis or they need to be shown to exist. Ideally, you should be able to outline the "form" of a proof of a given statement quickly, even if you can't fill in the details. – Jair Taylor May 18 '20 at 20:47
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    Aside from those issues, it helps to be good at manipulating inequalities and understanding basic set theory. I'd say, though, that mastering all of this is not necessarily a prerequisite, as your skills in these areas will grow as you progress through the material and work out problems. It certainly helps, though, to have a grader to critique your proofs in detail, so self-study could be tricky. – Jair Taylor May 18 '20 at 20:53
  • I had a bit of "trouble" assuming the induction principle for the model of $\mathbb N $. Also I never really liked the well-ordering principle (that every subset of natural numbers have a minimum): it seems un-natural to me to assume that the content of arbitrary subsets can be known (that is: in mathematics there is no potentiality, in the Aristotelian sense, everything is actual). However beyond that I dont remember something specially problematic. This is not directly related to real analysis but in most introductory courses of analysis you are exposed to the model of the natural numbers – Masacroso May 19 '20 at 06:24
  • As @JairTaylor said, make sure you fully understand reasoning in first-order logic. First you must have a complete grasp of the meaning of logical syntax, especially quantifiers. One way to quickly grasp it is via game semantics. After that, I personally recommend that you learn a Fitch-style deductive system such as this variant system, so that you can see precisely how all mathematical reasoning can be carried out, even if you do not actually write the steps down. – user21820 Jun 01 '20 at 05:51

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