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.