Final (condensed)

Question 1

Well Ordering Principle

Answer 1

Every non-empty subset of N has a smallest element

Question 2

Definition of Order

Answer 2

Let S be a set. An order on S is a relation denoted by <
with the following two properties:
1) If x∈S and y∈S, one and only one of the following statements are true: x < y, x = y, y < x. (Trichotomy)
2)If x,y,z∈S and x<y and y<z, then x<z. (Transitivity)

Question 3

Definition of a Field

Answer 3

A field F is a set equipped with two operations, addition and multiplication, which satisfy the field axioms

Question 4

Properties of an Ordered Field (other than trichotomy and transitivity)

Answer 4

1) If x,y∈F with x<y, then for all z∈F, x+z < y+z
2) If x,y∈F with x,y>0 then 0< x*y

Question 5

Definition of Supremum

Answer 5

Suppose S is an ordered set and E⊆S, and E is bounded above. Suppose there exists a B∈S with the following properties:
1) B is an upper bound of E
2) if a<B then a is not an upper bound of E.
Then B is the supremum or least upper bound.
Denoted B=sup(E)

Question 6

Definition of Infimum

Answer 6

Suppose S is an ordered set and E⊆S, and E is bounded above. Suppose there exists a A∈S with the following properties:
1) A is an lower bound of E
2) if A<b then b is not an lower bound of E.
Then A is the infimum, or greatest lower bound of E.
Denoted A = inf(E)

Question 7

Triangle Inequality

Answer 7

|x+y| <= |x| + |y|

Question 8

Reverse Triangle Inequality

Answer 8

||x|-|y|| <= |x-y|

Question 9

The Archemedian Principle

Answer 9

If x,y∈R and x>0 then there exists an n∈N s.t. n*x>y. Hence, for all x∈R, there exists an n∈N s.t. x<n

Question 10

Density of Q in R

Answer 10

For all x=/=y∈R, there exists a q∈Q s.t. x<q<y

Question 11

Definition of an Equilavence Relation

Answer 11

An equivalence relation is a relation which has the following properties:
1) Reflexive
2) Symmetric
3) Transitive

Question 12

Definition of a Surjection

Answer 12

Let A and B be two sets, and f:A->B be a function of A into B.
f:A->B is surjective if for each b∈B there exists an a∈A with f(a)=b

Question 13

Definition of an Injection

Answer 13

Let A and B be sets and f:A->B be a function of A into B.
f:A->B is injective if for a,b∈A with a=/=b, then f(a)=/=f(b)

Question 14

Definition of a Bijection

Answer 14

Let A and B be sets and f:A->B be a function of A into B. f is bijective if it is both injective and surjective

Question 15

Definition of a Convergent Sequence

Answer 15

A sequence {Xn}⊂R is said to converge to the real number a if for all ε>0 there exists an N∈N s.t. for all n>=N, |xn-a|<ε

Question 16

What two properties does a convergent sequence have?

Answer 16

Bounded and monotonic

Question 17

Bolzano-Weierstrauss Thm

Answer 17

Every bounded sequence in R has a convergent subsequence

Question 18

Cauchy Criteron

Answer 18

A sequence of {Xn}⊂R is cauchy iff {Xn} converges.

Question 19

Definition of a Cauchy Sequence

Answer 19

A sequence {Xn} is a cauchy sequence if for all ε>0 there exists an N∈N s.t. for all m,n>=N |Xn-Xm|<ε

Question 20

Definition of Limit Superior

Answer 20

If {Xn}⊂R then the limit superior of {Xn} is defined as lim n->∞ supXn = lim n->∞ (sup{Xk, k>=n} = inf(sup{Xk: k>=n})

Question 21

Definition of Limit Inferior

Answer 21

If {Xn}⊂R then the limit inferior of {Xn} is defined as lim n->∞ infXn = lim n->∞ (inf{Xk, k>=n} = sup(inf{Xk: k>=n})

Question 22

Definition of an Open Set

Answer 22

A set U⊆R is said to be open if for all x∈U, there exists an ε(x)>0 s.t. (x-ε,x+ε)⊆U

Question 23

Definition of a Compact Set

Answer 23

A subset K⊆R is compact if every open cover of K has a finite subcover

Question 24

Properties of compact sets

Answer 24

A compact subset of R is closed and bounded
Every infinite subset of K has a limit point in K.

Question 25

Properties of Limits

Answer 25

Let E⊆R, c a limit point of E, and f,g real valued functions with domain E. Suppose f(x)->L1, and g(x)->L2 as x->c, then: 1) f(x) + g(x) -> L1 + L2 as x->c 2) f(x) * g(x) -> L1 * L2 as x->C 3) If L2 =/= 0, f(x) / g(x) -> L1 / L2 as x->c

Question 26

Definition of a Limit

Answer 26

Let E⊆R and f a real valued function with domain E. Let c be a limit point of E. Then f has a limit of L at c if for all ε>0 there exists a δ(c,ε)>0 s.t. for all x∈E, if 0<|x-c|<δ, then |f(x)-L|<ε

Question 27

Definition of a Neighborhood

Answer 27

Let δ>0, c∈R. (c-δ,c+δ) is called a δ-neighborhood of c. (c-δ,c+d) \ {c} is called a deleted neighborhood of c.

Question 28

Squeeze Thm

Answer 28

Let E⊆R, c a limit point of E and f,g,h are real valued functions with domain E. Suppose f(x)<=g(x)<=h(x) on some deleted neighborhood of c. Then if f(x)->L and h(x) -> L as x->c then g(x)->L as x->c.

Question 29

What three properties are implied by the statement "f is continuous iff lim x-> c f(x) = f(c)"?

Answer 29

1) limx->c f(x) exists 2) f(x) exists at c 3) f(x) = f(c)

Question 30

Under what conditions is a function that is continuous guarateed to be uniformly continuous?

Answer 30

1) If f is continuous on a compact set K 2) If f is lipschitz continuous

Question 31

Definition of Lipschitz Continuity

Answer 31

Let E⊆R with f:E->R. f is Lipschitz continuous on E if there exists M>0 s.t. for all x,y∈E, |f(x)-f(y)| <= M|x-y|

Question 32

Limit definition of Derivative

Answer 32

Let I⊆R be an interval with f defined on I. For c∈I, the derivative of f at c is f'(c) = limx->c ( f(x) - f(c) ) / (x-c), provided the limit exists

Question 33

Properties of derivatives

Answer 33

1) (f + g)'(x) = f'(x) + g'(x) 2) (f*g)'(x) = f'(x)*g(x) + f(x)*g'(x) 3) (f / g)'(x) = ( f'(x)*g(x) - f(x)*g'(x) ) / [g(x)]^2

Question 34

Definition of Uniformly Continuous

Answer 34

Let E⊆R and f:E->R. f is said to be uniformly continuous if for all ε>0, there exists δ(ε)>0 s.t. for all x,y∈E with |x-y|<δ, |f(x)-f(y)|<ε