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…!!!