Lecture Notes 1 Flashcards
Ordered Set
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
Upper Bound
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
Lower Bound
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.
Least Upper Bound
Supremum
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.
Greatest Lower Bound
Infimum
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.
bounded
bounded above and bounded above. Or, you can draw a ball around the set.
Least Upper Bound Property
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.
Field
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
Ordered Field
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
Archimedean Property
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
Q is dense in R
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
inf{1/n: n in N} = 0
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.
Facts about the supremum
- if x is in R, then sup(x+A) = x+sup A
- if x>0, then sup(xA) = x(sup A)
- if x<0, then sup(xA) = x (inf A)
Facts about the infimum
- if x is in R, then inf(x+A) = x+ inf A
- if x>0, then inf(xA) = x(inf A)
- if x<0 then inf(xA) = x (sup A)
Supremum of an empty set
negative infinity
supremum of a set not bounded above
infinity
supremum of a set not bounded below
negative infinity
infimum of an empty set
infinity
Triangle Inequality
|x+y| less than or equal to |x|+|y|
Corollary to Triangle Inequality
- |(|x|-|y|)| less than or equal to |x-y|
2. |x-y| less than or equal to |x|+|y|
bounded (function)
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.
Cantor’s Theorem
- R is uncountable
2. the set (0,1] is uncountable
sequence
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}
bounded (sequence)
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
converge
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.
convergent
a sequence that converges
Convergent Sequence Facts
- a convergent sequence has a unique limit
2. convergent sequence is boundedd
monotone increasing
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.
monotone decreasing
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.
Facts about Monotone sequences
- bounded if and only if it is convergent
- monotone increasing and bounded: limit of x sub n as n goes to infinity is the supremum of the set
- 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
K-tail
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
subsequence
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}
Facts about convergent sequences
- if xn is a convergent sequence, any subsequence is also convergent and the limits are the same
Facts about limits of Sequences
- 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.
