MAT 3749 Introduction to Analysis

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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.

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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

• Q is a field.
• 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

• Determine which of the following sets are bounded
  above.
• Determine which of the following sets have a
  maximum element.

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Example 3 (a)
Analysis Solution
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Example 3 (b)
Analysis Solution
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Example 3 (c)
Analysis 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

Analysis Solution
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Example 5

• Determine the supremum of

Analysis Solution
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The Completeness Axiom