Test 1

Question 1

Upper Bound/Bounded Above

Chp 1

Definition

Answer 1

A number M is said to be an upper bound of E if x<=M for all x in E

Question 2

Lower Bound/Bounded Below

Chp 1

Definition

Answer 2

A number m is said to be a lower bound of E if m<=x for all x in E

Question 3

Maximum

Chp 1

Definition

Answer 3

If there exists a number M in E that is larger than every other member of E, then M is called the maximum of E

M=max(E)

Question 4

Minimum

Chp 1

Definition

Answer 4

If there exists a number m in E that is smaller than every other member of E, then m is called the minimum of E

m=min(E)

Question 5

Supremum

Chp 1

Definition

Answer 5

If E is bounded above and non-empty and M is the least upper bound, then M is the supremum of E

M=sup(E)

Question 6

Infimum

Chp 1

Definition

Answer 6

If E is bounded below and non-empty and m is the greatest lower bound, then m is the infimum of E

m=inf(E)

Question 7

The Completeness Axiom (Upper)

Chp 1

Answer 7

A non-empty set of Reals that is bounded above has a least upper bound

Hence, sup(E) exists

Question 8

The Completeness Axiom (Lower)

Chp 1

Answer 8

A non-empty set of Reals that is bounded below has a greatest lower bound

Hence, inf(E) exists

Question 9

Archimedean Properties

Chp 1

Theorem

Answer 9

1) The set of Natural numbers has no upper bound

2) No matter how large a Real x is, there is always a Natural n that is larger

3) No matter how large y>0 is and how small x>0 is, there exists a Natural n such that nx>y

4) No matter how small x>0 is, there exists a fraction (1/n)<x

Question 10

Well-Ordering Property

Chp 1

Theorem

Answer 10

Every non-empty subset of Naturals has a smallest element

Question 11

Dense Sets

Chp 1

Definition

Answer 11

A set of Reals, E, is said to be dense if every interval (a,b) contains a point of E

Question 12

What is the definition of Absolute Value?

Chp 1

Definition

Answer 12

|x| = x if x >= 0

|x| = -x if x < 0

Question 13

Absolute Value Properties

Chp 1

Theorem

Answer 13

For any x in R:

1) -|x| <= x <= |x|

2) |xy| = |x||y|

3) |x + y| <= |x| + |y|

4) |x| - |y| <= |x-y|
and
|x - y| >= |y| - |x|

Question 14

What is the distance between two real numbers?

Chp 1

Definition

Answer 14

d(x,y) = |x - y|

Question 15

What does a Sequence of Real numbers represent?

Chp 2

Definition

Answer 15

A sequence of Real numbers is a function f:N==>R

Question 16

What is Cantor’s Theorem?

Chp 2

Theorem

Answer 16

No interval (a,b) of Reals can be the range of some sequence

Question 17

Countable

Chp 2

Definition

Answer 17

A subset of Reals, S, is said to be countable if there is a sequence of Real numbers whose range is S

Question 18

Limit of a Sequence

Chp 2

Definition

Answer 18

If {Sn} is a sequence of Real numbers, we say {Sn} converges to a number L if for all epsilon>0, there exists N such that
|Sn-L|<epsilon>=N</epsilon>

Lim Sn = L

Question 19

Uniqueness of Limits

Chp 2

Theorem

Answer 19

If Lim Sn= L
and
Lim Sn=W

Then L=W

Question 20

Divergence

Chp 2

Definition

Answer 20

If {Sn} is a sequence of Reals, we say {Sn} diverges to (inf) if for every M there exists an N so that Sn>=M whenever n>=N

Lim Sn = (inf)

Question 21

What can you say about convergent sequences?

Chp 2

Theorem

Answer 21

They are bounded

Question 22

Lim cSn = ?

Chp 2

Theorem

Answer 22

c(Lim Sn)

Question 23

Lim (Sn+Tn) = ?

Chp 2

Theorem

Answer 23

Lim Sn + Lim Tn

Question 24

Lim (Sn/Tn) = ?

Chp 2

Theorem

Answer 24

(Lim Sn)/(Lim Tn)

Question 25

If {Sn} and {Tn} converge, and Sn<=Tn for all n, what can you say about the limits?

Chp 2

Theorem

Answer 25

Lim Sn <= Lim Tn

Question 26

If {Sn} converges and
a <= Sn <= b, what can you say about Lim Sn?

Chp 2

Theorem (Corollary)

Answer 26

a <= Lim Sn <= b

Question 27

If {Sn} and {Tn} converge
and
Lim Sn = Lim Tn for all n
and
Sn <= Xn <= Tn then…?

Chp 2

Squeeze Theorem

Answer 27

Lim Sn = Lim Xn = Lim Tn

Question 28

What is the limit of absolute values?

Chp 2

Theorem

Answer 28

The absolute value of the limit

Question 29

What is the Max/Min of a Limit?

Chp 2

Theorem (Corollary)

Answer 29

Lim Max{Sn,Tn}

Max{Lim Sn, Lim Tn}

Question 30

What is an increasing sequence?

Chp 2

Definition

Answer 30

S1 < S2 < … < Sn < …

Question 31

What is a non-decreasing sequence?

Chp 2

Definition

Answer 31

S1 <= S2 <= … <= Sn <= ..

Question 32

What is a decreasing sequence?

Chp 2

Definition

Answer 32

S1 > S2 > … > Sn > …

Question 33

What is a non-increasing sequence?

Chp 2

Definition

Answer 33

S1 >= S2 >= … >= Sn >= ..

Question 34

What is a monotonic sequence?

Chp 2

Definition

Answer 34

Any sequence that is non-increasing or non-decreasing (including increasing and decreasing)

Question 35

What is the Monotone Convergence Theorem?

Chp 2

Theorem

Answer 35

If {Sn} is a monotonic sequence, then {Sn} converges iff {Sn} is bounded

Question 36

If {Sn} is bounded and non-decreasing, what can you say?

Chp 2

Theorem

Answer 36

Since it is monotonic,
{Sn} ==> sup(Sn)

Question 37

If {Sn} is bounded and non-increasing, what can you say?

Chp 2

Theorem

Answer 37

Since it is monotonic,
{Sn} ==> inf(Sn)

Question 38

What is a subsequence?

Chp 2

Definition

Answer 38

If S1, S2, … is a sequence, then a subsequence exists S_n1, S_n2, … where n1 < n2 < …

Hence, n1, n2, … is a sequence of increasing Natural numbers

Question 39

Every sequence contains a _________ subsequence

Chp 2

Theorem

Answer 39

monotonic

Question 40

Bolzano-Weierstrass Theorem

Chp 2

Theorem

Answer 40

Every bounded sequence contains a convergent subsequence

Question 41

Cauchy Criterion

Chp 2

Theorem

Answer 41

A sequence {Sn} is convergent iff for each epsilon>0, there exists N such that |Sn - Sm|<epsilon, for all n,m>=N