Exam 2 Definitions and Theorems

R is a

Complete Archimedean ordered field

Field axioms

For additive and multiplicative, closure, commutativity, associativity, identity, and inverse & the distributive law

Order axioms

For all x, y, z \in R exactly one of the following holdv x=y, x<y, x>y; if x<y, y<z, then x<z; if x<y, then x+z<y+z; if x<y & z>0, then xz<yz

What other set is an ordered field?

Rational numbers

Theorem about relations of x, y, epsilon

Let x, y \in R s.t. \forall \epsilon > 0 in R, x<=y+\epsilon, then x<=y

Theorem about absolute value

Let x, y \in R, a>=0\in R. Then, |x|>=0, |x|<=a \iff -a<=x<=a, |xy|=|x||y|, |x+y|<=|x|+|y|

Triangle Inequality

\forall a,b,c\in R |a-c|<=|a-b|+|b-c|

Upper Bound for S\subset R

m s.t. m>=s \forall s\in S

Bounded Above set S

An upper bound exists for the set S

Lower Bound for S\subset R

m s.t. m<=s \forall s\in S

Bounded Below set S

An lower bound exists for the set S

Bounded set

A set that is bounded above and below

Maximum of S

An upper bound m\in S

Minimum of S

A lower bound m\in S

Supremum of S

(S\subset R and S is non-empty) The least upper bound of S if S is bounded above m=sup(s) \iff \forall s\in S, m>=s and if m>m’, then \exists s’\in S s.t. s’>m’

Infimum of S

(S\subset R and S is non-empty) The greatest lower bound of S if S is bounded above m=sup(s) \iff \forall s\in S, m<=s and if m’>m, then \exists s’\in S s.t. s’<m’

Complete Axiom

Every non-empty subset S of R that is bounded above has a least upper bound in R

What differentiates Q from R?

Q does not have the complete axiom

Theorem about sets of sums

Let A, B\subset R and define C={a+b:a\in A, b\in B}. If A and B have suprema, C has a supremum and sup(C)=sup(A)+sup(B)

Archimedean Property

N is unbounded in R

Equivalent statements to the Archimedean Property

For each z\in R, \exists n\in N s.t. n>z; for each x>0 and each y\in R, \exists n\in N s.t. n*x>y; for each x>0 in R \exists n\in N s.t. 0<1/n<x

Density of Q in R

If x, y\in R with x<y, then \exists r\in Q s.t. x<r<y

Lemma for the density of irrational numbers

If x≠0 is rational and y is irrational, then xy is irrational.

Theorem about irrational numbers

For x, y\in R, where x<y, then \exists irrational number w s.t. x<w<y

Sequence

A function whose domain is N (S: N→R)

A series (sn) converges to s if…

\forall \epsilon > 0, \exists m\in N s.t. \forall n>= m, |sn-s| < \epsilon

Limit of (sn)

s, where sn converges to s

Divergent sequence

A sequence that doesn’t converge to a real number

Bounded sequences (sn)

The set {sn:n\inN} is bounded (think (sn) can be put in an interval)

Theorem about bounded sequences

Every convergent sequence is bounded

Theorem about limits

If a sequence converges, then its limit is unique.

Theorem about algebraic relations

Spp (sn), (tn) convergent seq w/ sn→s and tn→t. Then, lim(sn+tn)=s+t, lim(ksn)=ks, lim(k+sn)=k+s \forall k\in R, lim(sntn)=st, lim(sn/tn)=s/t, provided tn, t≠0

Theorem on comparison of sequences

Spp (sn), (tn) convergent seq. w/ sn→s, tn→t. If sn<=tn \forall n, then s<=t

Corollary to comparison of series

(tn) convergent seq, tn→t w/ tn>=0, then tn>=0

Ratio test

For (sn) seq., sn>0 and (s(n+1)/sn)→L, if L<1, then sn→0

(sn) diverges to +\infty

\forall M\in R, \exists N \in N s.t. \forall n>=N, sn>M

Thm on divergent series

For (sn) seq. >0, lim sn=+\infty iff lim(1/sn)=0

(sn) is increasing

sn<=s(n+1) \forall n\in N

(sn) is decreasing

sn>=s(n+1) \forall n\in N

(sn) is monotone

It is increasing or decreasing

Monotone Convergence Theorem

A monotone sequence is convergent iff it is bounded

A seq. is Cauchy

if \forall \epsilon >0, \exists N\inN s.t. \forall m,n>=N, |sm-sn|<\epsilon

Theorem about Cauchy

A sequence is convergent iff it is Cauchy

Lemma about Cauchy

Every Cauchy sequence is bounded

Theorem connecting Cauchy to real numbers

The following are equivalent: The completeness axiom holds for R and every Cauchy sequence is convergent

Definition of the real numbers

Limits of Cauchy sequences of rational numbers

Lemmas on comparison of sup and inf

For non-empty A\in R that is bounded, infA<=supA. For B\subset A, supB<=supA and infB>=infA

Defn of limsup(xn)

lim(yn)=lim(n→\infty){sup{xk:k>=n}}

Corollary on limsup/liminf

liminf(xn)<=limsup(xn)

Theorem on convergence and limsup/liminf

(xn)→x \iff limsup(xn)=liminf(xn)=x

Theorem between convergence and Cauchy

A sequence of real numbers converges iff it is Cauchy

Theorem between Completeness Axiom and Cauchy

If every Cauchy sequence converges, Completeness Axiom holds

Lemma on non-unique decimal expansion

0.999..=1

Corollary on uniqueness of decimal expansions

If we forbid decimal expansions with all 9s, then decimal/Cauchy sequence expansions are unique

Theorem about the real numbers

The real numbers are uncountable

Definition of topology

Adding a structure of place/location to sets

Definition of neighborhood/\epsilon-neighborhood of x

Set of the form N(x, \epsilon):={y\in R: |x-y|<\epsilon} where x, \epsilon \in R and \epsilon>0

Radius of N(x, \epsilon)

\epsilon

Deleted \epsilon-neighborhood of x

N*(x, \epsilon):={y\in R: 0<|x-y|<\epsilon} (equivalent to N(x, \epsilon)\{x})

Interior Point of S\subset R

A point x\in R if \exists a nghd N of x s.t. N\subset S

Boundary Point of S\subset R

A point x\in R s.t. \forall nghd N of x, we have NnS ≠ \emptyset and Nn(R\S) ≠ \emptyset

Interior of S

The set of all interior points of S (int(S))

Boundary of S

The set of all boundary points of S (bd(S))

Lemma on interior/boundary points

Every point of S is either an interior point or boundary point

A set S\in R is open if:

One of the following, equivalent statements is true: int(S)=S; every point is an interior point; \forall x\in S, \exists \epsilon>0 s.t. N(x, \epsilon)\subset S

A set S\in R is closed if:

(R\S) is open

Lemma about open/closed sets

These traits are not mutually exclusive - a set can be neither open nor closed

Theorem about open sets

The union of any collection of open sets is open and the intersection of finitely many open sets is open

Corollary on closed sets

The intersection of any collection of closed sets is closed and the union of finitely many closed sets is closed

Accumulation Point of S

A point x\inR s.t. every deleted nghd of x contains a point in S

S’

The set of accumulation points of S

Isolated Point of S

x\inS and x\inS’

Closure of S\inR

SuS’ (denoted cl(S))

Theorem about closed sets

S closed iff S contains all accumulation points; cl(S) is closed; S closed iff S=cl(S); cl(S)=Su bd(S)

Open Cover of S\subsetR

Collection of open sets {U_i}_i\inI where Ui is open for every index i\inI s.t. S\subset\cup_i\inIU_i

The open cover {U_i} admits a finite subcover if

\exists some N\in\mathbb{N} s.t. S\subset\cup_i=1^N U_i

Compact set S\subset R

Every open cover of S admits a finite sub-cover

Theorem about compact sets

Compact iff bounded and closed