500 ALL

Question 1

Bolzano-Weierstrass Theorem (Star theorem)

Answer 1

Every Bounded real sequence has a convergent subsequence

Question 2

Monotone Convergence Theorem

Answer 2

Every Bounded monotone sequence converges

Question 3

Cauchy Sequences

Answer 3

if xn is cauchy, then for every e>0, there is an N (natural number) s.t. for all m,n >= N, then |xn - xm| < e (ALL Cauchy sequences are convergent!)

Question 4

Two-Sided Limit Convergence

Answer 4

Let I be an open interval, a is a point in I, and the function f be defined everywhere on I except possibly at a.
We say that f(x) converges to L as x->a IFF given e>0, there exists d> 0 s.t. for all points x in I, with 0< |x-a|

Question 5

Equality of limits of functions

Answer 5

Let I be an open interval, a is a point in I, and the functions f,g be defined everywhere on I except possibly at a.
If f(x) = g(x) for all points x in I except a, then lim f(x) = lim g(x)

Question 6

Sequential Characterization of Limits

Answer 6

Let I be an open interval, a is a point in I, and the function f be defined everywhere on I except possibly at a.
Then lim f(x) = L IFF lim f(xn) = L for all sequences {xn} with xn in I except at a, s.t. lim xn = a.

Question 7

Comparison Theorem of Limits of Sequences

Answer 7

If {xn} and {yn} converge, then if there exists an N natural number s.t. xn<=yn for all n >=N, then lim(xn) <= lim(yn)

Question 8

Bad influence Theorem on sequences

Answer 8

if lim(xn) -> 0 and yn is bounded, then lim(xn * yn) -> 0

Question 9

Squeeze Theorem on Sequences

Answer 9

If {xn} and {yn} both converge to a, and there exists a sequence {zn} s.t. xn < zn < yn, then {zn} -> a

Question 10

Limit of a Sequence

Answer 10

lim (xn) = a as n -> infinity if given e>0 there exists N a natural number s.t. n>=N implies |xn - a|<e

Question 11

How many limits can a sequence have

Answer 11

A limit can have at most one limit

Question 12

Definition of a Subsequence

Answer 12

A subsequence of the sequence {xn} is a sequence that’s obtained by taking a subset of the terms, like {xnk} with nk a natural number

Question 13

Convergence of subsequences

(for subsequences of a convergent sequence)

Answer 13

if lim(xn) -> a then every subsequence of {xn} also converges to a

Question 14

Definition of a Sequence

Answer 14

A sequence is a function whose domain is the Natural Numbers . IE it’s a function f: N->R where f(n) = xn is the nth term of the sequence

Question 15

Two sided Limit and One sided Limits equality

Answer 15

lim f(x) = L as x goes to a IFF lim f(x) x->a+ = lim f(x) x->a- = L

Question 16

Definition of a right sided limit

Answer 16

f(x) tends to L as x approaches a from the right IFF f is defined on an open interval whose left endpoint is a, and given any e>0 there exists d>0 s.t. for all a<x<a+d we have |f(x)-L|<e
lim f(x) as x->a+ = L

Question 17

Definition of a left sided limit

Answer 17

f(x) tends to L as x approaches a from the left IFF f is defined on an open interval whose right endpoint is a, and given any e>0 there exists d>0 s.t. for all a-d<x<a we have |f(x)-L|<e
lim f(x) as x->a- = L

Question 18

Comparison Theorem of limits of functions

Answer 18

Suppose f, g are functions defined on I{a} If lim f(x) and lim g(x) exist, and f(x) <= g(x) for all x in I{a}, then lim f(x) <= lim g(x)

Question 19

Squeeze Theorem on Functions

Answer 19

Suppose f, g, h defined on I{a} if f(x) <= g(x) <= h(x) for all x in I{a} and lim f(x) = lim h(x) = L, then lim g(x) =L

Question 20

Bad Influence Theorem on Functions

Answer 20

Let f, g be defined on I{a}. If g is bounded, so |g(x)| <= M for M a real number, and x in I{a} and if lim f(x) = 0, then lim [f(x) * g(x)] = 0

Question 21

limit of a sequence approaching supremum of a set

Answer 21

Let E be a subset of the real numbers. IF sup(E) < infinity, then there exist a sequence {xn} with elements in E st.t lim xn approaches supE

Question 22

Definition of Supremum

Answer 22

The supremum of a set E is the smallest upper bound of the set E.

Question 23

Supremum Approximation Property

Answer 23

If sup(E) is finite, then for all positive e, there exists an element a in set E s.t. a>sup(E)-e

Question 24

Completness Axiom

Answer 24

If E is a non-empty subset of the real numbers and if E is bounded above, then E has a finite supremum, sup(E) in the real numbers

Question 25

Archimedian Principle

Answer 25

Given any a, b real numbers with a>0, there exists n a natural number s.t. b<n*a

Question 26

Density of Rationals

Answer 26

The Rational numbers are dense. For any a,b rational numbers, with a<b, there exists a rational number q s.t. a<q<b

Question 27

Definition of Infimum

Answer 27

The infimum of a set E is the largest possible lower bound of the set E.

Question 28

Reflection Principle

Answer 28

Let E be a subset of the real numbers, and not an empty set. 1) E has a supremum iff -E has an infimum, and sup(E) = -inf(E) 2) E has an infimum iff -E has a supremum, and inf(E) is -sup(E)

Question 29

Monotone Property

Answer 29

Let A be a subset of B, both sets in real numbers, non-empty. Then, 1) if B has a supremum, then sup(A) <= sup(B) 2) if B has an infimum, then inf(B) <= inf(A)

Question 30

Well-Ordering Principle

Answer 30

If E is a non-empty subset of the NATURAL numbers, then E has a least element

Question 31

Induction Theorem

Answer 31

For every natural number n, supposed A(n) is a statement where A(1) is true, and for all n where A(n) is true, A(n+1) is also true

Question 32

Definition of a Set

Answer 32

A set is a collection of elements

Question 33

Definition of a Subset

Answer 33

A subset B of set A is if all of the elements in B are also in set A. If B is not equal to A, that is a proper subset

Question 34

A ∩ B

Answer 34

{x: x in A and x in B}

Question 35

A U B

Answer 35

{x: x in A or x in B}

Question 36

If E is a collection of sets, we have ∩A

Answer 36

{x: x in A for every A in E}

Question 37

If E is a collection of sets, we have UA

Answer 37

{x: x in A for some A in E}

Question 38

If B a subset of A, Complement of B in A

Answer 38

A\B, all of the elements of A not in B

Question 39

Definition of a function

Answer 39

A function f from set A to set B is a rule that assigns to each element of A exactly one element of B.

Question 40

Images

Answer 40

f(x) is called the image of x under f

Question 41

Onto/ Surjective

Answer 41

a function f:A->B is onto if for all elements y in B, there is an x in A where y is an element of f(x) (every point y in B has an associated point x in A)

Question 42

One-to-One / Injective

Answer 42

a function f:A-> is injective if f(x1)=f(x2) implies x1=x2 for all xn in A (Each x in A only has one corresponding point y in B

Question 43

Bijection

Answer 43

Both injective and surjective

Question 44

Definition of Function Inverses

Answer 44

for a function f:A->B has an inverse iff there exists a function g:B->A where g(f(x))=x for all x in A, and f(g(y)) = y for all y in B

Question 45

Uniqueness of inverses

Answer 45

If a function has an inverse, that inverse is unique

Question 46

definition of a two-sided limit as x goes to infinity

Answer 46

f(x)->L as x->inf iff there exists c>0 s.t. (c, inf) is in the domain of the function, and given any e>0, there exists an M >= c s.t for all x > M we have |f(x) - L| < e

Question 47

definition of two-sided limit as x goes to negative infinity

Answer 47

f(x)->L as x->-inf iff there exists c>0 s.t. (-inf, -c) is in the domain of the function, and given any e>0, there exists an M <= c s.t for all x < M we have |f(x) - L| < e

Question 48

Definition of a two-sided limit as f diverges to infinity

Answer 48

f diverges to infinity as x->a iff given M>0 there exists d>0 s.t. for all |x-a|<d we have f(x) > M

Question 49

Definition of a two-sided limit as f diverges to negative infinity

Answer 49

f diverges to negative infinity as x->a iff given M<0, there exists d>0 s.t. for all|x-a|<d we have f(x) <M