Trichotomy

For all a, b exactly one of these is true
a = b, a < b, or b < a
Applies to Q and R

UpperBound

Given A in R is a nonempty set, b is an upper bound for A if a <= b for all a in A

Supremum (Infimum)

s is a supremum (infimum) of A if s is an upperbound (lowerbound) for A and s<=b (s>=b) for all upperbounds (lowerbounds) b of A

Axiom of Completeness

If A of R is nonempty and bounded above, then sup(A) exists as a real number

Approximation Property

Given a nonempty set A of R such that s is an upper bound of A. Then s = sup(A) iff for all e > 0, there is an a in A such that se < a

Archimedian Property

Given x,y in R, both positive, there is an n in N such that nx > y

Density of Q in R

Given a < b where a,b in N, there is an r such that a < r < b and r in R

Equivalence Relation

A relation '~' on sets by A ~ B iff there is a bijection A > B ( 1to1 and Onto)

Countable

We say an infinite set A is countable if A = N_0 i.e. there is a bijection N > A : n > a_n

Contained Countable Sets

If A is contained by B and B is countable, then A is either finite or countable

Injection

1to1, two unique elements in the domain are never mapped to the same element in the codomain

Cantor's Theorem

For any set S, the size of S is less than the power set of S or the set of all subsets of S
S < P(S) where P(S) = {T contained in S}

Metric Space

A metric space (M,d) is a set M and a distance function d: M x M > R such that for all x,y, z in M:
1. d(x,x) = 0 and d(x,y) > 0 if x != y (Positivity)
2. d(x,y) = d(y,x) (Symmetry)
3. d(x,z) <= d(x,y) + d(y,z) (Triangle Ineq.)

Sequence

In a metric space (M,d), a sequence is an infinite list: x_1, x_2, x_3 ... i.e. a function
N > M: n > x_n
Various notations such as (x_n) from 1 to infinity, (x_n), or (x_1, x_2, x_3, ...)

Limit

A sequence (x_n) in a metric space (M,d) is said to have a limit L in M if for every e>0 there is an N in N (depending on e) such that n >= N ==> d(x_n, L) < e

Algebraic Properties of Limits of R Sequences

Let (a_n), (b_n) be convergent sequences of real numbers with limits a, b respectively
1. lim(c * a_n) = c * a for any c in R
2. lim(a_n + b_n) = a + b
3. lim(a_n * b_n) = a * b

Comparison Theorem

Let (a_n), (b_n) be convergent sequences of real numbers with limits a, b respectively. If a_n <= b_n for all n in N, then a <= b

Bounded

A sequence (x_n) in a metric space is bounded when there is a ball Br(x) = {m in M, d(m, x) < r} such that x_n is in Br(x) for all n in N

Convergent, Bounded Fact

Convergent sequences are bounded, but bounded sequences are not necessarily convergent.
Convergent ==> Bounded, Bounded =/=> Convergent

Increasing (Decreasing)

Given a sequence (a_n) of real numbers, we say (a_n) is increasing (decreasing) if a_n <= a_n+1 (a_n >= a_n+1) for all n in N
(a_n) is monotone if a_n is either increasing or decreasing

Monotone Convergence Theorem (MCT)

Given a bounded monotone sequence (a_n) of real numbers, there exists a limit
a = lim(a_n), i.e, a_n converges

Convergent Subsequences Lemma

Given a sequence (x_n) which is convergent , any subsequence (x_n1, x_n2, ... where n1<n2<n3...) is also convergent to the same limit

Cauchy

A sequence (x_n) in a metric space (M,d) is said to be Cauchy if for every e>0, there is some N in N such that d(x_m, x_n) < e whenever m,n >= N

Convergence and Cauchy Fact

A convergent sequence is cauchy.
Idea: Let x = lim(x_n). Then
d(x_m, x_n) <= d(x_m, x) + d(x_n, x) < e
for large m, n in N

BolzanoWeierstrass Theorem

Suppose (a_n) is a bounded sequence of real numbers. Then there exists a convergent subsequence

Nested Interval Property

Let I_1 = [a_1, b_1], I_2 = [a_2, b_2], ... be a sequence of closed finite intervals of real numbers. If the intervals are nested, i.e. I_1 contains I_2 contains I_3... then the intersection of all I_k is not empty.
Hence, I_n = [a_n, b_n] contains
[lim(a_n), lim(b_n)] for all n in N.
Therefore the intersection of all I_k contains [a,b] which is nonempty

Cauchy Criterion Theorem

A cauchy sequence in R is convergent
