Download presentation
Presentation is loading. Please wait.
1
Introduction to Analysis
J. Mercy Arokia Rani Assistant professor Department of Mathematics St. Joseph’s College(Autonomous) Trichy –
2
Math Party?
3
Actuarial Presnentation
Cambia/Regence 10/17 3pm OMH 126
4
Preview Field, Ordered Field Lower/Upper Bounds Supremum/ Infimum
5
References Section 1.1 Howland, Section 1.5
6
Introduction: A Story…
You are in a foreign country and want to buy….
7
Before going into that,.. We will briefly mention the field properties and that the real numbers R is a field.
8
Before going into that,.. We will briefly mention the field properties and that the real numbers R is a field.
9
Field
10
Real Numbers R The set R of Real Numbers is a field.
11
Example 1 Q is a field. Z is not a field. (Why?)
12
Ordered Field
13
Real Numbers R The set R of Real Numbers is an ordered field.
14
Example 2 C is a field but not an ordered field.
15
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)
16
Upper (Lower) Bounds Similar for bounded below, lower bound , and minimum element
17
Example 3 Determine which of the following sets are bounded above.
Determine which of the following sets have a maximum element.
18
Example 3 (a) Analysis Solution
19
Example 3 (b) Analysis Solution
20
Example 3 (c) Analysis Solution
21
Archimedean Property
22
Density Property
23
Least Upper Bounds Similar for greatest lower bound, and infimum
24
Equivalent Statement Similar for greatest lower bound, and infimum
25
Example 4 Show that Analysis Solution
26
Example 5 Determine the supremum of Analysis Solution
27
The Completeness Axiom
28
Thanking You…!!!
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.