Math 171 Notes

“The hardest thing about [the textbook] is pronouncing the names of its authors.” The course website is http://math.stanford.edu/~schoen/math171/. This is a course in mathematical analysis, so in addition to the content of the course, the class also emphasizes the art of writing proofs, and thus we will be nickled-and-dimed in our proofs. Additionally, this is a Writing in the Major course, so the writing assignment will require exposition as well as mathematical content. The first part of this class will be a quick treatment, ideally review, of properties of the real numbers: sequences, series, continuity, etc. In the textbook, this corresponds to chapters 1 to 6. A useful reference for this part of the class can be found at http://math.stanford.edu/~schoen/math171/simon.pdf, though it uses slightly different notation. One thing we could spend quite some time on, but aren’t going to, is the construction of the real numbers. It’s a bit tedious and not worth the time, but then there is a uniqueness result given these axioms (which is actually not that hard to prove). Instead, we’ll characterize the real numbers (denoted R) axiomatically; there are three sets of axioms. First are the algebraic axioms, which state that the real numbers form a field. F1. Both addition and multiplication are commutative: for all a, b ∈ R, a+ b = b+ a and ab = ba.