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