I am trying to teach myself real analysis, but I was wondering if this subject is just a collection of proofs of calculus theorems. Aside from set theory, I haven’t learned anything new. By that, I mean I haven’t come across any new theorems that I didn’t already know from calculus. Also, in calculus, there are many difficult problems, including challenging integrals and limits. However, the problems I have faced so far in real analysis were hard because of my proof-writing skills, not because they were difficult problems like integrals.
My second question is: If real analysis does not deal with hard integrals and if calculus books like Thomas' book don't have hard problems, then where does the insanely hard integrals that I see online came from ? something like nonelementary integrals or nonelementary function like $\operatorname{Li}(x)$. Where I can study them if not in real analysis or calculus ?
Best Answer
The beginning of Real Analysis certainly has significant overlap with the curriculum of most Calculus courses, though the emphasis is different. Whereas standard Calculus courses often emphasize the computation of things like limits, derivatives, and integrals, Real Analysis attempts to get students to think more rigorously about many of the concepts taught in Calculus.
"Aside from set theory, I haven't learned anything new" - the point is not to learn the results from Calculus all over again - you already learned the fundamental theorem of calculus, mean value theorem, intermediate value theorem, etc. when you took Calculus the first time. The point is to learn the proof techniques which are commonly used in Analysis. You should be fully comfortable being able to produce an $\epsilon -\delta$ proof all on your own to show things like convergence and continuity. You should feel confident in your understanding of how a metric endows a space with a topology and how this topology affects the properties of functions defined on that space. You should know how to work with different kinds of norms (e.g. supremum and operator norms) to show things like uniform convergence or boundedness. A standard first course in Real Analysis is not Calculus on steroids - it is not there to teach you how to compute more complicated integrals than the ones you see in Calculus - sorry but you won't find the logarithmic integral function $li(x)$ in most introductory Real Analysis books. It is there, instead, to give you the foundations you need to be successful in the study of mathematical objects defined in terms of limits.
"However, the problems I have faced so far in real analysis were hard because of my proof writing skills and not because they were difficult problems like integrals" - arguably, the exercises in Real Analysis have nothing to do with the integrals and everything to do with the proof-writing skills. You're trying to run before you can walk. You want to study special functions like $li(x)$, but the very definition of that function won't make any sense for ($x > 1$) if you haven't studied Complex analysis, and the techniques used in proving complex analysis results will be difficult to understand or employ if you haven't mastered them in real analysis.