Hi Jim, Sorry, I was rushed when I wrote my not-very-informative one-line recommendation. I used Arthur's book for a couple of years when I was teaching at Brandeis. The class was the "honors" intro to analysis course, with maybe a dozen students in it, and the book was pitched at a good level for them -- vastly more approachable than your "average MIT math major"-level worry. I felt like it had exactly the right approach to introducing the new material. including the "Hey, why do we need to think about this in the first place?" point of view that you wants in the introduction to the real numbers. My then-colleague Harry Tamvakis took an intense dislike to the book, on the other hand, because of its relaxed way of dealing with the epsilon-delta part of the material: Mattuck presents it as a one-variable definition, hiding epsilon with some English phrase like "as long as y is close enough to x." (He only gives epsilon an explicit name when needed for proofs like composition of continuous functions.) I thought this was a selling point myself. So YMMV. --Michael On Thu, May 28, 2009 at 10:34 PM, James Propp <jpropp@cs.uml.edu> wrote:
Michael Kleber suggested Mattuck's book. Mattuck is a good teacher, but I suspect that he wrote the book for MIT students, so I'm not sure how suitable it'd be for UMass Lowell students. UML does have some fine students, but the average UML math major is not as well- prepared as the average MIT math major.
I should also say that I'm NOT looking for a book that constructs the reals (via Dedekind cuts or Cantor sequences or Conway games). My view is that a student's first exposure to real analysis should be based on an axiomatic DESCRIPTION of the reals, not a formal CONSTRUCTION of them. My point of view might be expressed succinctly, with only a small amount of distortion, in the slogan "Why construct the reals when they already exist?"
Don't get me wrong; one of the formative experiences of my young mathematical career was getting a copy of Kirshner and Wilcox's "The Anatomy of Mathematics" as a gift when I was thirteen and devouring it from cover to cover. I loved the thrilling progression from Peano Postulates to the construction of the real numbers, with each successive number system incorporating the one before.
But most of my students won't become mathematicians, let alone logicians or model theorists (constructions are good for establishing relative consistency but it's a rare student who would worry about this). I want my students to become principled and savvy practitioners of real analysis who know what the main tools are and who know when caution is required in wielding them.
So I'm looking for a book by someone who shares that spirit.
On the other hand, I love a good (friendly) argument, so if any of you think I'm going in the wrong direction, please offer your counter-arguments!
Jim
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