Real Analysis
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last update: 18 July 2014
Lecture 24
A field is a set with operations
such that
(a) |
if then
|
(b) |
if then
|
(c) |
there exists such that
if then and
|
(d) |
if then there exists such that
and
|
(e) |
if then
|
(f) |
if then
|
(g) |
there exists such that
if then and
|
(h) |
if and then there exists
such that
and
|
(i) |
if then
|
Let be a field and let
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Add to each side and use (d) to get
|
Let be a field and let
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
By (d),
Add to each side and use (d) to get
|
Let be a field and let with
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
By (h),
Multiply each side by and use (h) and (g) to get
|
Let be a field and let
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
By (f),
where the last equality follows from Example 1. So, by (g),
Add to each side and use (d) and (c) to get
|
Let be a field and let
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
(-a)b+ab
=
(-a+a)b,by (f)
=
0·b,by (d)
=
0,by Example 1.
Add to each side and use (d) and (c) to get
|
Let be a field and let
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
|
Let be a set.
A total order on is a relation on such that
(a) |
if and
and then
|
(b) |
if and
and then
|
(b) |
if and
or then or
|
An ordered field is a field with a total order such that
(a) |
if and
then
|
(b) |
if and and
then
|
Let be an ordered field and let with
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Assume and
Then
by (OFb).
So by (Fd) and (Fc).
|
Let be an ordered field and let with
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Case 1:
Then by (OFb).
So by F Example 1.
Case 2:
Then by Example 1.
Then by Case 1.
So by F Example 6.
|
Let be an ordered field. Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
By (Fg),
|
Let be an ordered field. Let with
Show that
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Assume and
By Example 2,
So by (OFb).
So by (Fh) and F Example 1.
|
Let be an ordered field and let
Show that if and then
![](plus.gif) |
![](minus.gif) |
Proof. |
|
|
Let be an ordered field and let
Show that if then
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Assume
So and
Then, by Example 4, and
Thus, by (OFb)
Since then by (OFa).
So, by (OFb),
So, by (Fh),
So, by (OFa),
|
Let be an ordered field. Let
with and Then
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Assume and and
To show:
(a) |
If then
|
(b) |
If then
|
(a) Assume
Then and, since and
then
So
So
So
(b) Assume
Then
So
So
Since and then
Case 1:
Then and
So
So
So
Case 2:
Then and (since and
So
|
Notes and References
These are notes from a 2010 course on Real Analysis 620-295. This page comes from 100505Lect24.pdf and was given on 5 May 2010.
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