Lecture Notes 1

Question 1

Ordered Set

Answer 1

A set A, together with the relation < such that

i. for any x,y in A, exactly one of x<z
example: rational numbers, integers

Question 2

Upper Bound

Answer 2

Let E be in A, where A is an ordered set. If there is a b in A such that x is less than or equal to b for all x in E, then we say E is bounded above and b is the upper bound

Question 3

Lower Bound

Answer 3

Let E be in A, where A is an ordered set. If there exists a b in A such that x is greater than or equal to b for all x in E, then we say E is bounded below and b is a lower bound of E.

Question 4

Least Upper Bound

Supremum

Answer 4

If there exists an upper bound b0 of E such that whenever b is any upper boud for E we have b0 less than or equal to b, then b0 is called the least upper bound or the supremum of E.

Question 5

Greatest Lower Bound

Infimum

Answer 5

If there exists a lower bound b0 of E such that whenever B is any lower bound for E we have b0 greater than or equal to b, then b0 is called the greatest lower bound or the infimum of E.

Question 6

bounded

Answer 6

bounded above and bounded above. Or, you can draw a ball around the set.

Question 7

Least Upper Bound Property

Answer 7

An ordered set A has the least upper bound property if every nonempty subset E in A that is bounded above has a least upper bound, that is, sup E exists in A.

Question 8

Field

Answer 8

A set F is called a field if it has two operations defined on it, addition and multiplication, and if it satisfies the following axioms:
Addition:
1. if x is in F and y is in F, then x+y is in F
2. x+y = y+x
3. associativity of addition
4. zero element
5. negative element 
Multiplication:
1. if x is in F and y is in F, xy is in F
2. commutativity of multiplication
3. associativity of multiplication
4. element 1 such that 1x =x
5. reciprocals exist
Distributive law: x(y+z) = xy+xz

Question 9

Ordered Field

Answer 9

a field F is said to be an ordered field if F is also an ordered set such that
i. x0 and y>0 implies xy>0

Question 10

Archimedean Property

Answer 10

if x,y are in the Reals and x is greater than zero, then there is an N in the natural numbers such that Nx is greater than y

Question 11

Q is dense in R

Answer 11

if x,y are in the reals and x is less than y, then there exists an r in the rationals such that x<y

Question 12

inf{1/n: n in N} = 0

Answer 12

let A = {1/n: n in N} . Obviously A is nonempty. Furthermore, 1/n>0 and so 0 is a lower bound, and b := inf A exists. As 0 is a lower bound, then b is greater than or equal to zero. Now take an arbitrary a greater than zero. by the Archimedean property there exists an n such that na is greater than 1, or, in otherwords, a is greater than 1/n in A. Therefore a cannot be a lower bound for A. Hence b=0.

Question 13

Facts about the supremum

Answer 13

1. if x is in R, then sup(x+A) = x+sup A
2. if x>0, then sup(xA) = x(sup A)
3. if x<0, then sup(xA) = x (inf A)

Question 14

Facts about the infimum

Answer 14

1. if x is in R, then inf(x+A) = x+ inf A
2. if x>0, then inf(xA) = x(inf A)
3. if x<0 then inf(xA) = x (sup A)

Question 15

Supremum of an empty set

Answer 15

negative infinity

Question 16

supremum of a set not bounded above

Answer 16

infinity

Question 17

supremum of a set not bounded below

Answer 17

negative infinity

Question 18

infimum of an empty set

Answer 18

infinity

Question 19

Triangle Inequality

Answer 19

|x+y| less than or equal to |x|+|y|

Question 20

Corollary to Triangle Inequality

Answer 20

1. |(|x|-|y|)| less than or equal to |x-y|

2. |x-y| less than or equal to |x|+|y|

Question 21

bounded (function)

Answer 21

suppose F: D to Reals is a function. We say f is bounded if there exists a number M such that |f(x)| less than or equal to M for all x in D.

Question 22

Cantor’s Theorem

Answer 22

1. R is uncountable

2. the set (0,1] is uncountable

Question 23

sequence

Answer 23

A sequence of real numbers is a function x: Natural numbers to Real numbers. Instead of x(n) we usually denote the nth element in the sequence by x sub n. We use the notation {x sub n}

Question 24

bounded (sequence)

Answer 24

if there exists a B in the Reals such that |x sub n| is less than or equal to B for all n in the Natural numbers

Question 25

converge

Answer 25

A sequence is said to converge to a number x in the reals if for every epsilon greater than zero there exists an M in the Naturals such that |xsubn -x| is less than epsilon for all n greater than or equal to M. The number x is said to be the limit of the sequence.

Question 26

convergent

Answer 26

a sequence that converges

Question 27

Convergent Sequence Facts

Answer 27

1. a convergent sequence has a unique limit

2. convergent sequence is boundedd

Question 28

monotone increasing

Answer 28

a sequence is monotone increasing if x sub n is less than or equal to x sub (n+1) for all n in the Naturals.

Question 29

monotone decreasing

Answer 29

a sequence is monotone decreasing if x sub n is greater than or equal to x sub (n+1) for all n in the Naturals.

Question 30

Facts about Monotone sequences

Answer 30

1. bounded if and only if it is convergent
2. monotone increasing and bounded: limit of x sub n as n goes to infinity is the supremum of the set
3. monotone decreasing and bounded: limit of x sub n as n goes to infinity is the infimum of the set. 4. if S in Reals is a nonempty bounded set, then there exist monotone sequences xn and yn in S such that the supremum of S = lim xn n to infinity and the infimum of S = lim yn as n goes to infinity

Question 31

K-tail

Answer 31

the tail of the sequence starting at K+1
the sequence converges if and only if the K tail converges
the limit of the sequence and the limit of the K tail are the same

Question 32

subsequence

Answer 32

let xn be a sequence. let n sub i be a strictly increasing sequence of natural numbers. the sequence {x sub(n sub i)} is called a subsequence of {x sub n}

Question 33

Facts about convergent sequences

Answer 33

1. if xn is a convergent sequence, any subsequence is also convergent and the limits are the same

Question 34

Facts about limits of Sequences

Answer 34

1. squeeze lemma: let {an}, {bn} and {xn} be sequences such that an less than or equal to xn less than or equal to bn for all n in the Naturals.
   If {an} and {bn} converge to the same limit, then {xn} converges to that limit as well.