I am happy that you like the notes and more so that you're inspired to study this material. I have to admit though that I don't have any ready-to-offer self-study strategy for these notes. As you know, they began as supplementary lecture notes for a course I taught out of Spivak's *Calculus* text, and then they've been expanded upon in various ways since the course ended. I guess I want to be honest and say that for the typical student who would take such a course -- i.e., a bright undergraduate who is looking to get into real analysis from a quite modest starting point -- reading the notes alone would be a poor substitute for taking the class.

(I may not be admitting that much. For instance, compare to Rudin's *Principles of Mathematical Analysis*: this is a very popular text, but it is difficult to study on your own. There is a reason we have undergraduate math classes, after all.)

It is easier for me to answer in terms of the way the notes are written and what is not in them, so I'll try that.

1) The notes were written as a supplement to another text that is sparing in its actual text and has copious exercises, some of which are very challenging. So the notes go the other way: a lot more is proved in the text itself, and there are not nearly as many exercises. The advantage of that is that if what you want to see are more worked out examples of nontrivial arguments involving $\epsilon$, $\delta$, sups, Riemann sums, and so forth, there they are. The disadvantage is that there aren't nearly as many exercises as in most texts. Moreover the exercises appear intertextually only and tend to come up as direct responses to what was just done in the text rather than general reinforcement. Looking back at the text now, I find that there is exactly one exercise in Chapter 4 (Continuity and Limits). Uh oh! You'll almost certainly need to supplement with another text (e.g. Spivak!) here.

2) Spivak's text is wonderful at keeping things interesting. As mentioned above, he can keep things interesting with lively problems even when there is little to nothing going on in the text itself (e.g. Chapter 3 on functions). But once the text gets properly underway there are many wonderful digressions: for instance, he has an entire chapter each devoted to planetary motion, the irrationality of $\pi$ and the transcendence of $e$. Rudin's *Principles*, written in the 1950's, pioneered a much more streamlined approach. It gets plenty exciting eventually: Chapters 7 and 8 are set in Mount Orodruin, an amazingly gripping climax to all that has come before. Most analysis texts since Rudin have decided to either cover the same content as the first eight chapters of Rudin in twice the space or seem eager to get on to the graduate-level measure and function theory. The breadth and depth of undergraduate level real analysis that one finds in e.g. the *Calculus and Analysis* texts of Courant and John (several thousand pages worth of material!)
is mostly absent from modern texts.

Like Spivak, I wanted my notes to reflect the breadth and depth of undergraduate level real analysis. So it has a lot of excursions as well, but mostly different ones from Spivak. In particular I wanted to give a careful, lively treatment of certain topics that by now seem *never* to get a theoretical treatment in the undergraduate pure math curriculum: e.g. partial fractions, convexity, Newton's methods. So for me the most important chapters of the text are *Differential Miscellany*, *Integral Miscellany* and *Serial Miscellany*: i.e., three chapters full of (mostly) logically unnecessary diversions on derivatives, integrals and infinite series.

What does this mean for the self studier? I think it means: reading the notes carefully from start to finish is probably not a good idea, because things that are very ancillary are discussed along with things that are very important. As one example, I give a full blown treatment of the partial fractions decomposition, developing the portion of factorization theory in a univariate polynomial ring over a field necessary to do it, quite early on (Chapter 3). I put that in because in freshman calculus one *uses* the partial fractions decomposition to integrate rational functions whereas no calculus text I know of written in the last 50 years proves the partial fractions decomposition. Okay, but where is this used later in the text? Well, it *would* be used in Section 9.3.2 (integration of rational functions)...except that that section is currently blank. (Once you know the PFD, the discussion of integration of rational functions is exactly as in freshman calculus, except that giving a *general treatment* rather than what turns out to be a comprehensive-enough recipe necessitates a bunch of horrible notation, as I did cover a bit in my course. Also you have to pretend that you know how to factor polynomials over $\mathbb{R}$...which you actually don't without further theory e.g. due to Sturm.)

So you should not read carefully from cover to cover, but rather spend the most time on the material that is the most important. I think the text does a pretty good job at making the most important material easy to find: on a first reading I would concentrate on Chapters 4 (Limits and Continuity), 5 (Differentiation), 6 (Completeness), 8 (Integration), 10 (Sequences) and 11 (Series).

I hope that is at least somewhat helpful. Good luck.