Lecture Notes

Topics Covered (with key definitions & theorems):

(This is a work in progress, and organization will improve soon!)

1) Number systems: $\N$, $\Z$, $\mathbb Q$, $\R$, $\C$, others, & some of their properties

Archimedian Property

(Something we regrettably skipped: Dedekind's construction of $\R$ from $\mathbb Q$)

2) Max, min, upper bound, lower bound, sup, inf defined.

Completeness Axiom of $\R$: Every nonempty subset of $\R$ that's bounded from above has a least upper bound in $\R$ (+ analogous result for greatest lower bound)

Sequences and their limits

(epsilon & N definition of limit)

Some nice theorems about properties of limits, which we can use in lieu of the epsilon & N definition to quickly establish convergence (or non-convergence) . . . Cauchy sequences defined

Monotone sequences

Theorem: All bounded monotone sequences are convergent.

Theorem: As it turns out, Cauchy sequences are precisely the sequences that converge - i.e., we can use the Cauchy criterion as an equivalent definition of convergence. (Sometimes one definition is easier to work with than another in writing a proof, so this is good news).

lim inf, lim sup of a sequence (Thm: all bounded sequences have them)

Recursive sequences, & tricks for finding their limits, if extant (see Feb 4 note) (cobweb diagram)

Subsequences:

Every convergent sequence has a monotone subsequence

1-4) What are real numbers, anyway? Why do we need them? How can we rigorously define (ie, construct) them? What are some properties of $\R$ that other number systems don't have?

5) Sets, sequences, series… what's the difference?

What's a Cauchy Sequence?

6) How can you tell if a sequence converges (and, if it does, how can you tell what it converges to)?

7) How can you tell if a series converges (and, if it does, how can you tell what it converges to)?

What is radius of convergence?

What's a metric space?

Topological concepts are intuitive… until they're not. What are some caveats to watch out for?

Compact vs closed & bounded: when are these equivalent? When are they not equivalent?

What's so special about compact sets? (Ie, what are some theorems we proved about compact sets that won't hold for other kinds of sets?)

What's a continuous function? What conclusions can we make if we know a function is continuous? What conclusions might we be tempted to make, that actually aren't true?

==