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One to One
If whenever
or if
Sequence
A sequence is a function whose domain is the set of positive integers.
Supremum
Suppose is bounded above. A number is the supremum of if is an upper bound of and any number less the is not an upper bound of . We write = sup
Infimum
Suppose is bounded below. A number is the infimum of if is a lower bound of and any number greater the is not a lower bound of . We write = inf
Converges to a number L
A sequence {} converges to a number if
Z . The number is called the limit of the sequence and the notation
Bounded Below
The set is bounded below if there is a number such that . The number is called a lower bound of .
Bounded Above
The set is bounded above if there is a number such that . The number is called an upper bound of .
Bounded
The set is bounded if there is a number such that . The number is called a bound for .
Unbounded
A set is unbounded if for each number there is a point such that
Ordered Field
if a field and an ordered set with the following properties;
and
and
Interval
A set of real numbers is an interval iff contains at least 2 points and for any two points , every real number between and belong to as well.
Finite
A set is finite if it is empty or if its elements can be put in a one to one correspondence with the set {1,2,...,n} for some n.
Infinite
The set is infinite if it is not finite
Countably Infinite
The set is countably infinite if its elements can be put in a one to one correspondence with the set of positive integers
Countable
The set is countable if it is either finite or countably infinite.
Uncountable
The set is uncountable if it is not countable.
Cauchy sequence
A sequence is a Cauchy sequence if Z
Subsequence
Let be a squence and let be a strictly increasing sequence of positive integers.
The sequence is called a subsequence of
Onto
if
Order
<, on a set is a relation with the following properties;
Trichotomy, if , then only 1 of the following holds; or
Transitivity, , if and
Field
A non empty set, , with , with 2 operations, + and * w the following properties;
 Closure Commutative Associative Identity Inverse Distributive
Completeness Axiom
Each nonempty set of that is bounded above has a supremum
Interval
a Set, , of is an interval if it has the properties that if , and , then if
Triangle Inequality

Archimedean Principal
if , and
Well Ordering Principle
Every non empty set of Z+ has a least element
Reverse Triangle Inequality

Squeeze Theorem for Functions
Let I be an open interval containing point c and suppose are functions defined on I except pssibly at c.
Suppose \{c}
if then f has limit at c and
Balzano-Weierstrauss Theorem
Every bounded sequence has a convergent subsequence.
Subsequential Limit
The Subsequential limit of is any real number, x, such that there exists a subsequence of that converges to x
Differentiable at c
Let be an interval, let , and let . The function is differentiable at c provided that the limit

exists. The derivative of at c is the value of the limit and noted by
Intermediate Value Theorem
Suppose is continuous on . If
Extreme Value Theorem
if f:[a,b] \rightarrow \Re is continuous on then c,d
Uniformly continuous
Let be an interval. A function is uniformly continuous on if that satisfy .
Mean Value Theorem
if is continuous and differentiable on then there exists a point
Chain Rule
Let be an interval, ,
is defined on that contains
if is differentiable on , and is differentiable on then is differentiable on and
Intermediate Value Property
A function defined on an interval has the intermediate value property on if it satisfies the following condition: if and are distinct points in and is any number between and , then there exists a point between and such that
Limit Superior
if is bounded above then the limit superior is defined by
if not bounded above then the limit superior=
Limit Inferior
if is bounded below then the limit inferior is defined by
if not bounded below then the limit inferior = -
Right Hand Limit
is an open interval containing or is at endpoint. is a function defined on except maybe at
if or if is left endpoint, then has right hand limit L at c if
Left Hand Limit
is an open interval containing or is at endpoint. is a function defined on except maybe at
if or if is right endpoint, then has left hand limit L at c if
Limit of at c
has limit at if that satisfy

is an open interval containing point {}
Jump Discontinuity
has jump discontinuity at if the one sided limits exist but are not equal
Continuous at c
an open interval . is continuous at c if
that satisfy
Removable Discontinuity
at c if exists, but is either not defined or has a different value from the limit as x approaches c
Tagged Partition
A tagged partition of an interval consists of a partition
{ } of along with the set
{ } of points, known as tags, that satisfy for .
We will express a tagged partition of by
{ }
Norm of Partition, ||P||
The Norm of the partition P, ||P|| = max
If and are partitions of and is a refinement of
Riemann Sum
Let and let be a tagged partition of . The Riemann sum of associated with is defined by:

Partition, P
A Partition, P of interval is a finite set of points
{}
Riemann Integrable
A function is Riemann integrable on if there exists a number with the following property: for each , such that , for all tagged partitions of that satisfy
Riemann integral
The number L is called the Riemann integral of on and is denoted by the symbol , or simply .
Flashcard set info:
Author: Squiggleart
Main topic: Mathematics