MATH 4100/6100 (Azoff)
Real Analysis
  Fall 2005

This page was last updated on December 18.

Letter grades have been reported to the registrar.You can email me at azoff@math.uga.edu for your score on the final, class averages,etc.  You can also pick up your exam this week or next semester.

By popular request, solutions to the final exam have been posted at  4100fnlnotes.pdf.

Notes to the third hour exam are posted at 4100t3notes.pdf.


Here are links to pdf files containing notes to problems from previous assignments. 
Chapter 1 (Assignment 1)
Chapter 2 (Assignments 2&3)
Chapter 3 (Assignments 4&5)
Chapter 4 (Assignment 6)
Chapter 4 (Assignment 7)
Chapter 5 (Assignment 8)
Chapter 6 (Assignment 9)
Chapter 7 (Assignment 10)


Here is a link to a pdf file of P. Halmos' article "How to write mathematics".  It is both useful and humorous.

Here is a link to a pdf file of my MATH 3100 notes from last semester. The first chapter contains further discussion of the axioms of R, while the last chapter outlines several alternate methods of constructing R from Q.  Chapters 2 and 3 cover sequences and series.

Contents

  • Exams
  • Assignments
  • e-mail addresses and phone numbers of class members
  • Course syllabus


    • Exams

    There were three hour exams and a comprehensive final exam during the semester. 

    The first test covered Chapters 1 and 2; the median (including points for corrections), was 77% .

    The second hour test was held on Monday November 7.  It covered through Chapter 5; the median (including points for corrections) was 72%.

    The third hour test, covering through Section 7.25, was held on Tuesday, December 6.   Solutions are posted at 4100t3notes.pdf.  The median was 62%.

    The final was held 8-11 AM on Wednesday December 14.  Solutions are posted at 4100fnlnotes.pdf.  The median was 68%.

    Back to Contents


    Assignment Summary

    Topic
    Paragraphs
    Read For
    Discussion
    Problems
    Written Work
    # Due
    		 Problems
    		 Grad/Bonus
    Ordered Sets
    		 1.1-1.11
    		 M 22 Aug
    		 4
    		 1
    		 F 2 Sept
    		 1, 2, 3, 5, 9, 11, 12, 13, 17, 18
    		 A, B (see below), 6
    Fields
    		 1.12-1.20 W 24 Aug
    		 3
    Real & Complex #s
    		 1.21-1.32
    		 F 26 Aug
    		 8, 10
    Rk and Ck
    		 1.33-1.38
    		 W 31 Aug
    		 14, 15
    Countability
    		 2.1-2.14
    		F 2 Sept
    		 2
    		 2
    		 W 14 Sept
    		 1, 3, 4, 5, 6, 7, 8, 9abc, 10abc,11cd
    		2, 9def, 11ab
    Open/Closed Sets
    		 2.15-2.30
    		 F 9 Sept
    		 11e
    Compactness
    		 2.31-2.42
    		 W 14 Sept
    		 16
    		 3
    		 W 21 Sept
    		 10d, 12, 13, 14, 15, 19, 20, 22
    		 17, 24, 25, 29
    Connectedness
    		 2.43-2.47
    		 F 16 Sept
    		 18, 21
    Convergent Seqs.
    		 3.1-3.4
    		 M 19 Sept
    		 3
    		 4
    		 M 3 Oct
    		 1, 2, 4, 5, 20, 21, 23
    		 16
    Subsequences
    		 3.5-3.14
    		 F 23 Sept
    		 22
    Upper/Lower Limits
    		 3.15-3.20
    		 W 28 Sept
    		 5
    Series
    		 3.21-3.29
    		 M 3 Oct
    		 6c
    		 5
    		 M 10 Oct
    		 6, 7, 8, 9, 10, 13, 19
    		 11, 14abc
    Root & Ratio Tests
    		 3.30-3.40
    		 W 5 Oct
    		 9d
    Abs Convergence
    		 3.41-3.55
    		 F 7 Oct
    		 15
    Limits
    		 4.1-4.5
    		 M 10 Oct
    		 Examples in R
    		 6
    		 M 17 Oct
    		 1, 2, 3, 4, C (below)
    		 5,6,7
    Continuity
    		 4.6-4.12
    		 W 12 Oct
    		 vs sequences
    Continuity & Compactness
    		 4.13-4.21
    		 F 14 Oct
    		 10
    Continuity & Connectedness
    		 4.22-4.27
    		 M 17 Oct
    		 convexity
    		 7
    		 M 24 Oct
    		 8,9,11a,12,14,16,17,20
    		11b,18,23
    Functions on R
    		 4.28-4.34
    		 W 19 Oct
    		 21, 24
    Derivatives
    		 5.1-5.6
    		 F 21 Oct
    		 2, 13
    		 8
    		 W 2 Nov
    		 1,3,5,7,8,9,11,12,25
    		 4,14,26,27
    Mean Value Thm
    		 5.7-5.13
    		 M 24 Oct
    		 4, 22
    Taylor's Thm
    		5.14-5.19
    		 W 26 Oct
    		 20
    Upper/Lower Sums
    		 6.1-6.2
    		 F 28 Oct
    		 4
    		 9
    		 F 18 Nov
    		 2,4,5,8,15a,1,3,18
    		 6,10,13
    Integrability
    		 6.3-6.7
    		 M 31 Oct
    		 5
    Integrable classes
    		 6.8-6.9
    		 W 2 Nov
    Integral properties 6.12-6.13
    		 W 9 Nov
    		 7
    Fundamental Th
    		 6.20-6.22
    		 F 11 Nov
    		 15
    Stieltjes Integral
    		6.10,6.14-6.19
    		 M 14 Nov
    		 9
    Arc Length
    		 6.26-6.27
    		 W 16 Nov
    		 19
    Iterated Limits
    		 7.1-7.6
    		 F 18 Nov
    		 log and exp
    		 10
    		 F 2 Dec
    		 1,2,3,5,7,9,16,18
    		 11,15
    Uniform Conv
    		 7.7-7.10
    		 M 21 Nov
    		 8
    Continuity
    		 7.11-7.18
    		 M 28 Nov
    		 4
    Equicontinuity
    		 7.19-7.25
    		 W 30 Nov
    		 19
    Power Series
    		 8.1,8.6
    		 F 2 Dec
    		 1
    		 X*
    		 Th 8 Dec*
    		 1,4,5,8
    		 10
    Transcendental Fns
    		 8.7
    		 M 5 Dec
    		 7
    Test 3
    		 6.1-7.25
    		 Tu 6 Dec




    Mop up
    		 1.1-8.7
    		 W 7 Dec





    * Assignment X will not be collected.

    Supplementary Problems

    A.  Let F be an ordered field and suppose a,b in F+.  Prove the following.
            (i)  inf F+ = 0.     
            (ii)  If  a< b, then an<bn for each positive integer n.
           (iii)   If an<bn for some positive integer n, then a<b.

    B.  Suppose F is an ordered field, while n is a positive integer.  Prove the following.
           (i)  If 0<x<1, then  (1+x)n-1≤nx(1+x)n-1≤n2nx.
           (ii)  inf {bn | b in F, b>1} =1
           (iii)  sup {bn | b in F, 0<b<1} =1

    C.  Redo Problems 3 and 4b on the first hour test by first showing that the function f mapping X to the R by f(x):=d(x,b) is continuous and then using results from Chapter 4.
     
     

    Back to Contents  

    Phone Numbers and e-mail Addresses.

    Go By Last Name email Phone
    Bob Allen
    		 boballen@uga.edu 706-254-5504
    Stephen Bismarck
    		 sbis1@uga.edu 706-316-3638
    Ryan Byrd
    		 gyphdid@uga.edu 706-254-1375
    Eli Cleveland
    		 goober@uga.edu 706-548-7945
    Robert
    		 Darrith
    		 rdarrith@uga.edu 912-996-5338
    Jennifer Ellis
    		 jbelton@uga.edu 731-695-4337
    Erica Fields
    		 erica7@uga.edu 404-375-1670
    Samantha Gradolf
    		 sammie64@uga.edu 678-427-4101
    Brad Holderfield
    		 bradhold@uga.edu 678-475-4643
    Erin Horst
    		 emhorst@uga.edu 503-799-9176
    Aja Johnson
    		 ajaj@uga.edu 706-621-2489
    Ken Knox
    		 knoxk@uga.edu 256-653-7773
    Hyeonmi Lee
    		hmdoban@uga.edu 706-389-6347
    Soo-Jin Lee
    		 sjjh0314@uga.edu 706-255-6439
    Velma Moon
    		 velmavm@uga.edu 770-873-2312
    John Moragne
    		 johnbm1@uga.edu 706-355-5437
    David Prager
    		 pragerdj@uga.edu 706-542-2593
    Amelia Reeves
    		 reeves@math.uga.edu 912-977-1945
    Floyd Rinehart
    		 marker@uga.edu 706-534-1401
    Tim Rondeau
    		 trondeau@uga.edu 770-596-0272
    Darren
    		 Rose
    		 dlrose@uga.edu 404-374-8134
    Brandon Samples
    		 bsamples@uga.edu 770-403-5254
    Sheree Sharpe
    		 ssharpe@uga.edu
    Sally Shaul
    		 sshaul@uga.edu 678-315-9321
    Jaehong Shin
    		 jhshin@uga.edu 706-255-6439
    Catherine Ulrich
    		 culrich@uga.edu 706-353-1746
    Dominic Valentino
    		 dev@uga.edu 770-337-0120
    Justin Vastola
    		 jostaman@uga.edu 678-910-2672
    George Vulov
    Stephen
    		 Winburn
    		 steve30@uga.edu 706-714-5975

    Back to Contents

    MATH 4100/6100 (Azoff)
      Real Analysis
    Course Syllabus

    Web Page  http://www.math.uga.edu/~azoff/courses/4100.html 



    Call Numbers
    		 01-606 for MATH 4100 and 21-607 for MATH 6100



    Prerequisites
    		 Sequences and Series (MATH 3100) and Introduction to Higher Mathematics (MATH 3200) 



    Time & Place  9:05 - 9:55 AM  MWF
    		 302 Boyd



    Objective Learning precise real variable concepts and using them in rigorous proofs.



    Text Principles of Mathematical Analysis,  3rd Edition, by Walter Rudin, McGraw Hill Publishing Company, 1976;
    ISBN 0-07-054-235-X.



    Topics Real and Complex Number Systems
    Basic Topology
    Numerical Sequences and Series
    Continuity
    Differentiation
    Riemann and Riemann-Stieltjes Integrals
    Sequences and Series of Functions
    		1.5 weeks
    2.5 weeks
    2 weeks 
    2 weeks 
    2 weeks 
    2 weeks 
    2 weeks 



    Grading Homework 
    Hour Tests (3 @ 100 pts) 
    Final Exam    		 100 points 
    300 points 
    200 points




    		Homework will be collected weekly;  no late work will be accepted.   Graduate students must work at least one bonus problem on each assignment.  The final exam is scheduled for 8 - 11 AM on Wednesday December 14;  it will be comprehensive. 



    Instructor  E. Azoff
       e-mail 
       Phone 
       Office     		 azoff@math.uga.edu
    542-2608 
    443 Boyd 
      Office
      Hours    		 8:30 - 9:30 AM on Tuesdays and Thursdays
    2:30 - 3:30 PM on Mondays, Wednesdays, and Fridays
    No Office Hours on
    Oct 4, 5, 13, 18, 19, 25, or 26


    Assignment for Chapter 1

    Topic
    Paragraphs
    Read For
    Discussion
    Problems
    Written Work
    # Due
    		 Problems
    		 Grad/Bonus
    Ordered Sets
    		 1.1-1.11
    		 M 22 Aug
    		 4
    		 1
    		 W
    31
    Aug
    		 1, 2, 3, 5, 9, 11, 12, 13, 17, 18
    		 A, B (see below), 6
    Fields
    		 1.12-1.20 W 24 Aug
    		 3
    Real & Complex #s
    		 1.21-1.32
    		 F 26 Aug
    		 8, 10
    Rk and Ck
    		 1.33-1.38
    		 M 29 Aug
    		 14, 15

    Supplementary Problems for Assignment 1

    A.  Let F be an ordered field and suppose a,b in F+.  Prove the following.
            (i)  inf F+ = 0.     
            (ii)  If  a< b, then an<bn for each positive integer n.
           (iii)   If an<bn for some positive integer n, then a<b.

    B.  Suppose F is an ordered field, while n is a positive integer.  Prove the following.
           (i)  If 0<x<1, then  (1+x)n-1<=nx(1+x)n-1<=n2nx.
           (ii)  inf {bn | b in F, b>1} =1
           (iii)  sup {b^n | b in F, 0<b<1} =1

    Back to Contents