Well, I have a large collection of analysis in pdf format - I've been reading wikis, and
Mathematical Analyis, Apostol whenever things get hairy.
The course book is
Vector Calculus, Linear Algebra, and Differential Forms by the Hubbards; the authors are a married couple who, oddly enough, write math books together.
The class I'm taking covers linear algebra, multi-variable calculus, and real analysis over two semesters. Right now we're studying row reduction, and how row reduction can be used as a proof method. It's interesting because, from my understanding, it's uncommon to use row reduction in proofs. Apparently, this can be used to prove that only square matrices are invertible.
Some proofs I've learned:
Proof of the intermediate value theorem.
Prove, using continuity and the Bolzano Weiestrass theorem that a compact, real valued continuous function has a supremum
M, and is continuous at a point
a such that
f(a) = M.
http://en.wikipedia.org/wiki/Bolzano–Weierstrass_theorem
Prove that a function
f is continuous at a point
x_0 if and only if for all sequences
x_i converging to
x_0, the limit
i approaches infinity for
f(x_i) = f(x_0) is true.
If we want to do this, perhaps we could narrow the collaboration down to common proofs, and syntax/semantics of analysis. What proofs do your books cover? Perhaps I could obtain a pdf copy of the books others are using.
My background is in computer science. I've committed quite a bit of time to getting scheme with scmutils up and running. It's used with this book:
http://mitpress.mit.edu/SICM/
I also have prolog, but haven't spent as much time with it. I'd like to use prolog to develop a functional interpretation of analysis; I'm not a fan of how everything is so rigorous, yet everything is typically presented in not so rigorous notation.
