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48 Cards in this Set
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WellOrdering Principle of N

Every nonempty subset of N has a least element.

Algebraic properties of R

(A1)a+b=b+a
(A2)(a+b)+c=a+(b+c) (A3)0+a=a (A4)a+(a)=0 (M1)a*b=b*a (M2)(a*b)*c=a*(b*c) (M3)1*a=a (M4)a*(1/a)=1 (D)a*(b+c)=(a*b)+(a*c) 
The order properties of R
There is a nonempty subset P of R, called the positive real numbers, with these properties: 
(1)If a,b belong to P, then A+B belongs to P.
(2)If a,b belong to P, then ab belongs to P. (3)Trichotomy Property 
Trichotomy Property

If a belongs to R, then one of the following holds:
a∈P, a=0, a∈P 
The notion of inequality btw 2 real numbers

Let a,b be elements of R.
(a)If ab∈P, then a>b or b<a (b)If ab∈P∐{0}, then a≥b or b≤a 
4 Results of ordering properties of R

1)If a>b and b>c then a>c
2)If a>b then a+b>b+c 3)If a>b and c>0 then ca>cb. 4)If a>b and c<0 then ca<cb. 
The absolute value of a real number a is

. a if a>0
. 0 is a=0 . a if a<0 
Triangle Inequality

If a,b∈R, then ∣a+b∣≤∣a∣+∣b∣

Two variations of the triangle inequality

. ∣∣a∣∣b∣∣≤∣ab∣
. ∣ab∣≤∣a∣+∣b∣ 
Additional(4) Properties of absolute value

(a) ∣ab∣=∣a∣∣b∣ ∀ a,b∈R
(b) ∣a∣^2 = a^2 ∀ a∈R (c) If c≥0, then ∣a∣≤c iff c≤a≤c (d) ∣a∣≤a≤∣a∣ 
Let a∈R and ε>0. Then the εneighborhood of a

is the set Vε(a):={x∈R:∣xa∣<ε}

For a∈R, the statement that x belongs to Vε(a) is equivalent to

ε<xa<ε and aε<x<a+ε

Let a∈R. If x belongs to the neighborhood Vε(a) for every ε>0,

then x=a

Let S be a nonempty subset of R.
The set S is said to be bounded above if 
∃ a number u∈R s.t. s≤u ∀s∈S.
u is called an upper bound of S. 
Let S be a nonempty subset of R.
The set S is said to be bounded below if 
∃ a number w∈R s.t. w≤s ∀s∈S.
w is called a lower bound of S. 
Let S be a nonempty subset of R.
A set is said to be bounded if 
it is both bounded above and bounded below.

Let S be a nonempty subset of R.
A set is said to be unbounded if 
it is not bounded.

Let S be a nonempty subset of R.
If S is bounded above, then a number u is said to be a supremum (a least upper bound) of S if it 
(1)u is an upper bound of S, and
(2)if v is any upper bound of S, then u≤v sup S 
Let S be a nonempty subset of R.
If S is bounded below, then a number w is said to be an infimum (a greatest lower bound) of S if it 
(1)w is a lower bound of S, and
(2)if t is any lower bound of S, then t≤w inf S 
Archimedean Property

IF x∈R, then ∃ nₓ∈N s.t. x<nₓ

If a,b∈R satisfy a<b, then the open interval set is

(a,b):={x∈R:a<x<b}

If a,b∈R satisfy a≤b, then the closed interval set is

[a,b]:={x∈R:a≤x≤b}

Characterization Theorem
(intervals) 
If S is a subset of R that contains at least two points and has the property,
(1)if x,y∈S and x<y, then [x,y]⊆S, then S is an interval. 
Nested Intervals
A sequence of intervals I(n), n∈N is nested if 
I(1)⊇I(2)⊇....I(n)⊇I(n+1)⊇...

Nested Intervals Property

If Iₓ=[aₓ,bₓ], x∈N is a nested sequence of closed bounded intervals, then ∃ a number έ∈R s.t έ∈Iₓ for all x∈N

A sequence of real numbers is a function

defined on the set N={1,2,...._ of natural numbers whose range is contained in the set R of real numbers.

Subsequence definition
Let X=x(n) be a sequence of real numbers and let n₁<n₂<....<nₓ<.... be a strictly increasing sequence of natural numbers. 
Then the sequence X'=x(nₓ) given by {x(n₁),...,x(nₓ),..} is a subsequence of X.

Subsequences of convergent sequences also converge to the same limit

If a sequence X=x(n) of real numbers converges to a real number x, then any subsequence X'=x(nₓ) of X also converges to x.

Divergence Criteria
If a sequence X=x(n) of real numbers has either of these properties then X is divergent 
(1)X has 2 convergent subsequences X'=x(nₓ) and X"=x(rₓ) whose limits are not equal.
(2)X is unbounded. 
Monotone Subsequence Theorem

If X=x(n) is a sequence of real numbers, then there is a subsequence of X that is monotone.

BolzanoWeierstrass Corollary

A bounded sequence of real numbers has a convergent subsequence.

Nested Intervals Proof

 intervals are nested, then I(n)⊆I(1) for all n∈N so a(n)≤b(n) for all n∈N
 there is a nonempty set {a(n):n∈N} that is bounded above  let ξ be its supremum  then a(n)≤ξ for all n∈N  claim ξ≤b(n) for all n∈N Case1: If n≤k, with I(n)⊇I(k), then a(k)≤b(k)≤b(n) Case2: If k<n, with I(k)⊇I(n), then a(k)≤a(n)≤b(n) therefore, a(k)≤b(n) for all k, so b(n) is upper bound for set {a(k):k∈N} ξ≤b(n) for each n∈N a(k)≤ξ≤b(n) for all n which means ξ∈I(n) for all n∈N 
Prove
If a>b and b>c then a>c 
If ab∈P and bc∈P then (ab)+(bc)=(ac)∈P
therefore a>c 
Prove
If a>b then a+b>b+c 
If ab∈P then (a+c)(b+c)=(ab)∈P

Prove
If a>b and c>0 then ca>cb. 
If ab∈P and c∈P then cacb=c(ab)∈P
therefore ca>cb when c>0 
Prove
If a>b and c<0 then ca<cb. 
If c<0, then c∈P so that cbca=(c)(ab)∈P
therefore cb>ca when c<0 
Completeness Property of R

Every nonempty set of real numbers that has an upper bound also has a supremum in R.

Limit of a sequence

A sequence X=x(n) in R is said to converge to x∈R, or x is a limit of x(n), if for every ε>0 ∃ a natural number K(ε) s.t. for n≥K(ε), the terms x(n) satisfy ∣x(n)x∣<ε

Uniqueness of limits

A sequence in R can have at most 1 limit

Prove Uniqueness of limits

Suppose lim(x(n))=X' and X". For each ε>0 ∃ K' s.t. ∣x(n)X'∣<ε/2 for all n≥K' and ∃ K" s.t. ∣x(n)X"∣<ε/2 for all n≥K". Let K be the larger of K' and K". Then for n≥K, with the Triangle Inequality,
∣X'X"∣=∣X'x(n)+x(n)X"∣ ≤∣X'x(n)∣+∣x(n)X"∣ <ε/2+ε/2=ε Since ε>0 then X'X"=0 
Limit Theorems: X+Y
Let X=x(n) and Y=y(n) be sequences of real numbers that converge to x and y 
sequence X+Y converges to x+y

Limit Theorems: XY
Let X=x(n) and Y=y(n) be sequences of real numbers that converge to x and y 
sequence XY converges to xy

Limit Theorems: X*Y
Let X=x(n) and Y=y(n) be sequences of real numbers that converge to x and y 
sequence X*Y converges to xy

Limit Theorems: cX
Let X=x(n) be sequence of real numbers that converge to x and let c∈R 
sequence cX converges to cx

Squeeze Theorem

Suppose X=x(n), Y=y(n) and Z=z(n) are sequences of real numbers s.t. x(n)≤y(n)≤z(n) for all n∈N and the lim(x(n))=lim(z(n)).
Then Y=y(n) is convergent and lim(x(n))=lim(y(n))=lim(z(n)) 
Monotone Sequence

X=x(n) is a sequence then X is monotone if it either increases or decreases

BolzanoWeierstrass Theorem

If an infinite set S is contained in a finite interval [a,b] then the set S has at least 1 limit point A in the interval [a,b]

Limit Point

A number x s.t. for all ε>0, ∃ a member of set y, different from x, s.t. ∣yx∣<ε
