Download 1.1 The Real Numbers

MAT 3749
Introduction to Analysis
Section 1.1
The Real Numbers
http://myhome.spu.edu/lauw
Math Party?
Actuarial Presnentation
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Cambia/Regence
10/17
3pm
OMH 126
Preview
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Field, Ordered Field
Lower/Upper Bounds
Supremum/ Infimum
References
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Section 1.1
Howland, Section 1.5
Introduction: A Story…
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You are in a foreign country and want to
buy….
$2.00
$3.00
$5.00
$9.00
$10.00
$2.79
$3.00
Before going into that,..
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We will briefly mention the field
properties and that the real numbers R is
a field.
Before going into that,..

We will briefly mention the field
properties and that the real numbers R is
a field.
Field
Real Numbers R
The set R of Real Numbers is a field.
Example 1
(a)
(b)
Q is a field.
Z is not a field.
(Why?)
Ordered Field
Real Numbers R
The set R of Real Numbers is an ordered
field.
Example 2
C is a field but not an ordered field.
Move On…
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More properties of R.
First important milestone of an analysis
class – supremum / infimum
(Allow us to prove results such as
Intermediate Value Theorem)
Upper (Lower) Bounds
Similar for bounded below, lower bound , and
minimum element
Example 3
1.
2.
Determine which of the following sets
are bounded above.
Determine which of the following sets
have a maximum element.
Example 3 (a)
E1  0, 2
Analysis
Solution
Example 3 (b)
1

E2   n  Z  
n

Analysis
Solution
Example 3 (c)
E3  N
Analysis
Solution
Archimedean Property
Density Property
Least Upper Bounds
Similar for greatest lower bound, and infimum
Equivalent Statement
Similar for greatest lower bound, and infimum
Example 4
Show that sup 0, 2  2
Analysis
Solution
Example 5
Determine the supremum of
Analysis
Solution
 1
E  1  2 n 
 n




The Completeness Axiom