# Yet other introductions to analysis

If memory serves, Bryant's "Yet Another Introduction to Analysis" is the recommended text at Baruch. But there are other texts that can compete with Bryant as introductions to the subject. I want to look briefly at a couple.

One problem in recommending a text on analysis is the plethora of excellent treatments out there. One is spoilt for choice. I can easily recommend twenty or thirty excellent introductions to the subject, all published during the last decade or so. The one I grew up on was Bartle and Sherbert's "Introduction to Real Analysis," which is a fine text. But I am not recommending it here. The reader I have in mind is a clod who somehow got through the calculus sequence and needs to learn some analysis because, dammit, he's determined to be a quant. But learning analysis shouldn't be too painful -- not like root canal work without novocaine.

Two texts come to mind that rival Bryant in user-friendliness:
1) A First Course in Mathematical Analysis, by J.C. Burkill, published by Cambridge, and

All three of the authors -- Bryant, Burkill, and Howie -- are English academics and their books are pitched at first-year undergraduates who've done "A" levels in math at school (i.e., they're comfortable with differentiating and integrating functions but don't know how the subject hangs together).

Burkill is an old classic, originally published in 1961, which has gone through several reprints on account of its popularity. It's a slender book of about 180 pages, written in a semi-conversational style, which introduces the ideas of real numbers, sequences, continuity, differentiation, and integration in short and separate chapters. The reals are introduced in terms of Dedekind cuts (rather than the l.u.b. property or as Cauchy sequences of rationals). And I note that Burkill doesn't even bother defining Cauchy sequences but restricts the discussion to showing that every monotone bounded sequence has a limit. This is a very basic book.

Howie is relatively recent, published in 2001, and again has been reprinted several times. It has 276 pages, comparable to the 290 pages of Bryant. The pages, however, are larger, and the typesetting more attractive. The book works through the usual topics of reals, sequences, continuity, differentiation, integration. However, Howie goes further than Bryant by introducing the idea of uniform convergence and applying the notion to power series -- Bryant discusses power series but elides over uniform convergence. In addition, Howie introduces the trig functions by defining them as power series and then deriving their properties -- that's the way it should be done rather than drawing pictures, which is Bryant's approach. Howie's final (but brief) chapter has a discussion of Stirling's formula and the construction of a continuous but non-differentiable function. Again, this is important: ideally a text should show how analysis can reveal the properties of special functions -- log, exp, trig functions, hyperbolic functions, and so on.

However, it's not the discussion of slightly more advanced topics that makes the book attractive. It is pedagogically well-written. The author has taught elementary analysis for years upon years and this experience makes itself felt in the layout, organisation and discussion of topics. There are numerous remarks explaining what is really going on; there are adequate worked examples; and all the problems have worked solutions at the back.

#### dstefan

##### Baruch MFE Director
Interesting and timely. We will look at these texts and, possibly, choose another text for our undergraduate course.

#### Ergodic

From a pedagogical standpoint, Abbott's Understanding Analysis is--according to this MAA reviewer--so good that it's dangerous.

Its treatment of power series/elementary functions is probably not as good as the ones BBW recommended, though; it works best in the hands of the intellectually curious student who wants to understand why many of the subtle results of the subject have to be stated and proved explicitly. But in that realm it's probably among the best modern texts.

Just my $0.02. Also curious if anyone else has encountered this book. The reader I have in mind is a clod who somehow got through the calculus sequence and needs to learn some analysis because, dammit, he's determined to be a quant. I don't suppose any of those "clods" will be reading your recommendations? And I note that Burkill doesn't even bother defining Cauchy sequences... :-k Forgive my skepticism, but I don't see how you can discuss and use the Cauchy Criteria for convergence of series (shrinking tails) without the basic concept of Cauchy sequences. Nor can you have any meaningful discussion of Bolzano-Weierstrass type processes... #### DanM ##### Math Student #### bigbadwolf From a pedagogical standpoint, Abbott's Understanding Analysis is--according to this MAA reviewer--so good that it's dangerous. Its treatment of power series/elementary functions is probably not as good as the ones BBW recommended, though; it works best in the hands of the intellectually curious student who wants to understand why many of the subtle results of the subject have to be stated and proved explicitly. But in that realm it's probably among the best modern texts. Just my$0.02. Also curious if anyone else has encountered this book.

I have a copy, which I was looking at prior to reading your post. It's a fine text (among other fine texts) but. The style and choice of topics indicate the book is meant for motivated math majors. Not for the clod who's taken basic calc and is dreaming of becoming a quant.

One other book I might recommend -- at least for supplementary reading, if not as a core text -- is David Bressoud's "A Radical Approach to Real Analysis." This is a leisurely and historically grounded introduction to the subject (read, however, the second Amazon review, by one Viktor Blasjo, who takes apart some mistakes in the historical account made by Bressoud). I have the first edition but the second came out four years ago with 56 additional pages.

---------- Post added at 02:53 PM ---------- Previous post was at 02:50 PM ----------

Has anyone used/read Real Analysis by N.L. Carothers?

Seen it but don't have it. It's at the level of a grad course in real analysis.

#### DanM

##### Math Student
Seen it but don't have it. It's at the level of a grad course in real analysis.

Interesting. I'm going to be using it in my 4th year Real Analysis course.

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