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