How We Got from There to Here: A Story of Real Analysis
From Number to Cantor's Theorem, this book brings you on a journey of the development of mathematical analysis. Several important stops along the way include Taylor Series, the Bolazano-Weierstrass Theorem, and Cauchy Sequences, I cannot think of any notable omissions along the road.
Spot on, though the text is built on problems that leave a lot of work for the reader.
The approach the authors take is essentially timeless, in that it brings us to modern analysis. I imagine fifty years from now we could still look at this book as a very good exposition of how we got to where we are in twentieth century mathematics, and that will still be quite relevant for our moderately advanced undergraduates and casual mathematically curious students.
There are times when the prosaic nature of the narrative is a little strained, but the intention is to make the text more accessible. It is not a serious detraction, nor does it significantly get in the way of the text's movement. In the final analysis, it is a pleasure to read and the text moves logically.
Very strong. The authors have a genuine interest in the story they weave and keep the tale together throughout the text.
The text builds naturally through the history of mathematical analysis, so modularity is not itself a strong objective. With that said, you can reasonable go to any section and learn from it, but this text intentionally and appropriately is intended to be taken more as a sequential whole than in unrelated parts.
Time is a natural progression -- the time of mathematics development and how results build from one generation of mathematicians to the next -- and this text flows naturally through this development.
The text is well put together and easy to use. The links in the PDF are useful.
I observed no significant grammatical issues
This is exactly the approach to deliver cultural literacy to the budding mathematically inquisitive student. This text contains much of the lore of mathematical analysis that is appropriate for our undergraduate students.
This is a delightful text, and exactly how we should develop undergraduate analysis to our mathematical studies students. There are several classic results, but more importantly it shows how analysis develops, and hence why modern analysis has the character it does. This is how we should prepare our students for advanced undergraduate or graduate analysis, giving them a perspective of mathematics. For example, instead of taking an informal aside to explain the countability of the rationals, this is done naturally through the text (Problem 201).
The text leaves many details, including most proofs, as exercises. If you adopt this text you will probably want to take many of these as illustrative examples in your lecture development. Hence, there would be a significant amount of preparatory work to deliver this material, but it gives a very good framework to base your class on.