MAT 3749 Introduction to Analysis Section 1.1 The Real Numbers .
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Transcript of MAT 3749 Introduction to Analysis Section 1.1 The Real Numbers .
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MAT 3749Introduction to Analysis
Section 1.1
The Real Numbers
http://myhome.spu.edu/lauw
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Math Party?
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Actuarial Presnentation
Cambia/Regence 10/17 3pm OMH 126
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Preview
Field, Ordered Field Lower/Upper Bounds Supremum/ Infimum
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References
Section 1.1 Howland, Section 1.5
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Introduction: A Story…
You are in a foreign country and want to buy….
$2.79 $3.00
$3.00
$5.00
$2.00
$10.00
$9.00
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Before going into that,..
We will briefly mention the field properties and that the real numbers R is a field.
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Before going into that,..
We will briefly mention the field properties and that the real numbers R is a field.
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Field
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Real Numbers R
The set R of Real Numbers is a field.
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Example 1
(a) Q is a field.
(b) Z is not a field.
(Why?)
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Ordered Field
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Real Numbers R
The set R of Real Numbers is an ordered field.
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Example 2
C is a field but not an ordered field.
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Move On…
More properties of R. First important milestone of an analysis
class – supremum / infimum (Allow us to prove results such as
Intermediate Value Theorem)
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Upper (Lower) Bounds
Similar for bounded below, lower bound , and minimum element
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Example 3
1. Determine which of the following sets are bounded above.
2. Determine which of the following sets have a maximum element.
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Example 3 (a)
1 0,2E Analysis Solution
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Example 3 (b)
2
1E n Z
n
Analysis Solution
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Example 3 (c)
3E NAnalysis Solution
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Archimedean Property
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Density Property
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Least Upper Bounds
Similar for greatest lower bound, and infimum
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Equivalent Statement
Similar for greatest lower bound, and infimum
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Example 4
Show that sup 0,2 2
Analysis Solution
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Example 5
Determine the supremum of 2
11E nn
Analysis Solution
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The Completeness Axiom