MAT 3749Introduction to Analysis

Section 1.1

The Real Numbers

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Preview

Field, Ordered Field Lower/Upper Bounds Supremum/ Infimum

References

Section 1.1 Howland, Section 1.5

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

Before going into that,..

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) Q is a field.

(b) 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…

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. Determine which of the following sets are bounded above.

2. Determine which of the following sets have a maximum element.

Example 3 (a)

1 0,2E Analysis Solution

Example 3 (b)

2

1E n Z

n

Analysis Solution

Example 3 (c)

3E NAnalysis 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 2

11E nn

Analysis Solution

The Completeness Axiom