syllabus
From the handout (attached to the home page for the course):
The following topics will form the core of the course:
- Foundations: The necessary logic, proof techniques and set theory. Most likely this will be your first exposure to the ``axiomatic approach" that characterises modern mathematics.
- Real Numbers: A descriptive (and axiomatic) approach to the real numbers. What {\em is} that number line thingy that you thought you'd understood for over a decade?
- Convergence and Continuity: Sequences, limits and continuous functions. All familiar from calculus, but here we'll go much more deeply into their workings.
- Differentiation: Again you'll recognise the big picture from calculus, but I hope by this point in the course you'll be eager to revisit the topics with your newly acquired mathematical perspective.
This core corresponds, with some additions and omissions to be announced as we go, to the first six chapters of Lay's "Analysis with an Introduction to Proof". This is the course text; you will need access to a copy.
We'll start with a refresher of what you know from calculus and, more importantly, look at what your calculus course glossed over in order to press on to the important applicable results. This will take about a week and we'll need two weeks or so for each of the above four bullets. This leaves us some time to either follow interesting tangents or press on into further topics.
Interesting tangents could include a more considered look at the foundations of math (logic and set theory), a constructive approach to the real numbers (we'd build them out of the integers, via the rationals), more topological results, and more pathological fractal curves. The two most natural further topics are integration (including the fundamental theorem of calculus) and power series.
