Real Analysis Midterm

Field Axioms

upper and lower bounds

Supremum Theorem

LUBP

Archimedean Property

Archimedean Property effect

Decimal Expansion

Cardinality

finite, infinite, countable, uncountable, countably infinite

prime factorization

ennumeration

metric space definition

open ball definition

open set definition

metric space open qualities

closed subset definition

De Morgan’s laws

metric space closed properties

convergence definition

bounded sequence

order preserved by limits

monotone sequence

monotone convergence theorem

subsequence

convergence and subsequences

Bolzano-Weierstrass theorem

Cauchy sequence

convergence and cauchy

complete metric space

cauchy and boundedness

cauchy and subsequence convergence

partial sums and infinite series

convergence of a series

geometric series convergence

partial sums and convergnece

absolute convergence

comparison lemma

ratio test

root test

lemma necessary for convergence

if Ean converges, then an → 0

proof: if Ean converges, the the sequence of partial sums Sn = En_k=1 (ak) n>=1 converges to some S e R. Then an = Sn - Sn-1 → S - S = 0

closed subsets and convergence

A subset of F of a metric space is closed IFF every convergent sequence of points in H has its limits in F

completness of R

diagonalization