6/18/11
Well-Ordering Principle
Every non-empty subset of the set of
natural numbers has a least element.
Well-Ordering
Principle
This follows from the Peanos fifth
postulate.
Well-Ordering Principle
Proof:
By contradiction
Suppose that there is a non-emp

6/18/11
Peanos Postulates
Axiomatic Development
of the
Real Number System
P1: 1 is an element of the set X.
P2: For each element
there is a
unique
called the successor of n.
P3: For each
P4: If
are distinct, then
P5: Let
and
If for all
then
Construction o

7/3/11
Quiz
1. When is a sequence increasing?
Parabola
Decreasing?
2. What is a lower bound of a sequence?
Upper bound?
3. Show that
is convergent.
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Standard Equations
Example
Vertex:
vertex
p
distance of vertex from focus
and directrix
Example
Opening u

6/23/11
Sequences
Sequences
of Real Numbers
List of numbers in a definite order
Function whose domain is the set of natural
numbers
Behavior of the sequence as n tends to
infinity
Examples
Limit of a Sequence
The sequence
we write
has the limit L and
i

6/30/11
Quiz
1. What is a sequence?
2. Give an expression for the n-th term of
the sequence whose first four terms are
Monotone
Convergence
Theorem
3. Use the definition to show that
Convergent Sequences
Convergent Sequences
Monotone Convergence
Theorem
E

6/30/11
Locus of Equations
Functions and
Relations
The locus of an equation is a curve
containing those points, whose
coordinates satisfy the equation.
A point lies on a curve if and only if its
coordinates satisfy the equation of the
curve.
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Exam