Test 1 Flashcards
A set A ⊆ R is bounded above if
There exists a number b ∈ R
such that a ≤ b for all a ∈ A. The number b is called an upper bound for A.
Axiom of Completeness
Every nonempty set of real numbers that is bounded above has a least upper bound.
set A is bounded below if
There exists a lower bound l ∈ R
satisfying l ≤ a for every a ∈ A.
A real number s is the least upper bound for a set A ⊆ R if it meets the following two criteria:
(i) s is an upper bound for A;
(ii) if b is any upper bound for A, then s ≤ b
complement
Given A ⊆ R, the complement of A, written Ac, refers to the set of all elements of R not in A. Thus, for A ⊆ R,
Ac = {x ∈ R : x /∈ A}.
A function from A to B is
a rule or mapping that takes each element x ∈ A and associates with it a single element of B. In this case, we write f : A → B. Given an element x ∈ A, the expression f(x) is used to represent the element of B associated with x by f. The set A is called the domain of f. The range of f is not necessarily equal to B but refers to the subset of B given by {y ∈ B : y = f(x) for some x ∈ A}.
Maximum and minimum of a set
A real number a0 is a maximum of the set A if a0 is an element of A and a0 ≥ a for all a ∈ A. Similarly, a number a1 is a minimum of A if a1 ∈ A and a1 ≤ a for every a ∈ A
Countable and uncountable
A set A is countable if N ∼ A.
An infinite set that is not countable is called an uncountable set
The set A has the same cardinality as B
(this is pretty much density)
If there exists f : A → B that is 1–1 and onto. In this case, we write A ∼ B
The power set P(A) refers to
The collection of all subsets of A
What is a sequence
A sequence is a function whose domain is N
Convergence of a Sequence
A sequence (an) converges to a real number a if, for every positive number ε, there exists an N ∈ N such that whenever n ≥ N it follows that
|an − a| < ε
ε-neighborhood
Given a real number a ∈ R and a positive number ε > 0, the set
V(a) = {x ∈ R : |x − a| < ε}
is called the ε-neighborhood of a
monotone, increasing, decreas-
ing
A sequence (an) is increasing if an ≤ an+1 for all n ∈ N and decreasing if an ≥ an+1 for all n ∈ N.
A sequence is monotone if it is either increasing or decreasing.
Convergence of a Series
say that the series bn converges to B if the sequence of partial sums (sm) converges to B.
subsequence
Let (an) be a sequence of real numbers, and let n1 < n2 < n3 < n4 < n5… be an increasing sequence of natural numbers. Then the sequence
(an1 , an2 , an3 , an4 , an5…)
is called a subsequence of (an) and is denoted by (ank), where k ∈ N indexes
the subsequence.
Cauchy Criterion
A sequence (an) is called a Cauchy sequence if, for every ε > 0, there exists an N ∈ N such that whenever m, n ≥ N it follows that |an − am| < ε
Nested Interval Property
For each n ∈ N, assume we are given a closed interval In = [an, bn] = {x ∈ R : an ≤ x ≤ bn}.
Assume also that each In contains I(n+1)
Then, the resulting nested sequence of closed intervals has a nonempty intersection
Monotone Convergence Theorem
If a sequence is monotone and bounded, then it converges
Bolzano-Weierstrass Theorem
Every bounded sequence
contains a convergent subsequence.
Archimedean Property
(i) Given any number x ∈ R,
there exists an n ∈ N satisfying n > x.
(ii) Given any real number y > 0, there exists an n ∈ N satisfying 1/n < y
Density of Q in R
For every two real numbers a and b
with a<b, there exists a rational number r satisfying a<r<b
For every y ∈ R, there exists a
sequence of rational numbers that converges to y
R is
uncountable
Uniqueness of Limits
The limit of a sequence, when it
exists, must be unique
