This page is for my notes and the course material as I understand it, although it may contain some errors as it remains a constant work in progress. There are also questions after the notes to help improve my understanding.
Prop: if r=c/d≠0 is a rational number and r satisfies the equation c_n*x^n+…+c_1*x+c_0=0 with c_i∈Z, c_n≠0, c_0≠0, then d|c_n and c|c_0.
Corollary: if c_n=1, then d=1, meaning r is an integer.
Least Upper Bound of a set S = sup(S) = min{α|α is an upper bound of S}
Greatest Lower Bound of S = inf(S)
Completeness Axiom: if S is bounded from above, then sup(S) exists. Corollary: if S is bounded from below, then inf(S) exists.
Archimedian Property: if a,b>0, then ∃n∈N such that na>b.