Lecture Notes

1. Numbers, Sets, and Sequences

Rational Zeros Theorem. For polynomials of the form c_nxⁿ + … + c₀ = 0 , where each coefficient is an integer, then the only rational solutions have the form $\frac{c}{d}$ where c divides c_n and d divides c₀; rational root r must divide c₀.

The maximum of a set S is the largest element in the set.
The minimum is the smallest element in the set.
The $\inf$ of S is the greatest lower bound.
The $\sup$ of S is the smallest upper bound.
S is bounded if $\forall$s $\in$ S, s$\leq$M for some M $\in$ $\reals$
Completeness Axiom. If S is a nonempty bounded set in $\reals$, then $\inf$ S and $\sup$ S exist.
Archimedean Property. If a, b $\gt$ 0, then $\exists$n such that na $\gt$ b.

A sequence (s_n) is a function mapping from $\N$ to $\R$. It converges to s if $\forall$ $\epsilon$ > 0 there exists N such that N > n $\implies$ |(s_n)-s| < $\epsilon$
In other words, $\lim$(s_n) = s

Important limit theorems include:
$\lim$(s_n)(t_n) = ($\lim$(s_n))($\lim$(t_n))
$\lim$(s_n)+(t_n) = ($\lim$(s_n)) + ($\lim$(t_n))
$\lim$($\frac{1}{n^p}$) = 0 for p > 0
$\lim$ n^(1/n) = 1

A subsequence (s_n(k)) of (s_n) is a sequence that is a subset of the elements in the original sequence with relative order preserved.
Bolzano-Weierstrass Theorem. Every bounded sequence has a convergent subsequence, having some subsequential limit.

Given any (s_n) and let S be the set of subsequential limits of (s_n). Define:
$\lim$ $\sup$ (s_n) = $\lim\limits_{N \to \infin}$ $\sup${(s_n): n > N} = $\sup$ S
$\lim$ $\inf$ (s_n) = $\lim\limits_{N \to \infin}$ $\inf${(s_n): n > N} = $\inf$ S

$\lim$ $\inf$ |s_n+1| / |s_n| $\leq$ $\lim$ $\inf$ |s_n|^(1/n) $\leq$ $\lim$ $\sup$ |s_n|^(1/n) $\leq$ $\lim$ $\sup$ |s_n+1| / |s_n|

2. Topology