Elementary Real Analysis
Fall `07 - Humke`s Version
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- Text: Real Analysis, A First Course, 2^{nd} Edition by
Russell Gordon
- TGWSUF: Paul Humke Office: OMH303 ext. 3410
- Hotline: 645-6440
- Office Hours: MWF 2:00-3:00, TTh 4:00-5:00
- Class Folder I'd like you to maintain
a loose leaf notebook
divided into 3 sections: Homework, Definitions, Theorems. What is to go into each section is
briefly described below.
- Homework - 100 pts.- Daily assignments are due 2 class periods after they are assigned.
Working together to solve the problems is encouraged big time!!,
but you must write up your own solutions.
- Definitions -- I'd like you to keep a complete, dictionary
(ie. an alphabetized list) of the definitions we learn in the course.
This should be ``word processed''. Flash cards with the word to be defined on one
side and the definition on the other works for me. Give it a try.
- Theorems -- In this section of the Class Folder I'd like you to
list the important theorems (facts) of the course organized according to
the chapters in the book. Under the statement of each theorem, you should list
from 1 to 4 key ideas of the proof of that theorem. This too should be ``word processed''.
You might flashisize these theorem statements too.
- Midterms - 200 pts.- There will be 2 midterm exams each
worth 100pts.
- Final - 200 pts.- The Final, celebrated on Monday, December 17
at 2:00-4:00, will be comprehensive. (Don't forget to study!!)
- Some Suggestions - 10,000 Lira-
- I suggest making flash cards
for your definitions and theorem statements. Put the word to be defined (or
theorem statement) on the front and the definition (or proof outline)
on the back. I know this sounds crazy, but these are the basics and if you know
the basics you'll be in good shape.
- Get into an ERA Study Group and make sure a
real smart person is in that group with you. Working on these ideas together is fun and helps make
your study time more useful. We have a huge ERA class, so helping each other is a MUST!
- See Humke regularly. Hey, he can help, he gets
paid huge bucks to help you,
and he gets lonely sitting in his office all by himself. Help the poor guy out and
get a few ERA tips while your at it. What could be better!!
- ERA is a coiurse about ideas, and mastering the ideas of week 3
depends on your having mastered the ideas of week 2. ERA can overwhelm you
if you don't keep on top of things. If you do get behind, (see above).
- Some Cool Links
Below is a tentative list of Assignments, ERA Words, ERA Facts and More
Chapter 1
- Section 1.1 What is a Real Number
- Definitions
- set
- natural number
- rational number
- real number
- field
- ordered set
- ordered field
- Theorems
- Gordon Problems 1,2,3,4,6,7,10,12 (Assigned 9-7-05; Due 9-12-05)
Chapter 1
- Section 1.2 Absolute Value, et. al.
- Definitions
- |x| (absolute value of x)
- max (min) of a set
- interval
- Arithmetic Mean
- geometric Mean
- E is dense in R
- Theorems
- Triangle Inequality
- Reverse Triangle Inequality
- Geometric Sum Theorem
- Arthmetic Mean >= Geometric Mean
- Mathematical Induction
- Cauchy-Schwartz Inequality
- Gordon Problems (Assigned Due )
- Read Appendix C,
- Prove 1+2+...+n=n(n+1)/2.
- Prove 1^{2}+2^{2}+...+n^{2}=n(n+1)(2n+1)/6.
- 1,2,4,6,9,15,19,32
Chapter 1
- Section 1.3 The Completeness Axiom
- Definitions
- E is bounded (above)(below)
- a=inf(E)
- b=sup(E)
- Completeness Axiom
- Theorems
- Archimedian Property
- Q is dense in R
- Gordon Problems 2,5,9,10,12,26,28
(Assigned ; Due )
Chapter 1
- Section 1.4 Countable and Uncountable Sets
- Definitions
- finite set
- infinite set
- aleph-naught
- countable set
- uncountable set
- f:A->B is one-to-one
- f:A->B is onto
- c
- Theorems
- Cantor-Schroder-Bernstein Theorem
- Countable unions of countable sets are countable
- card(Q)=aleph-naught
- c is not aleph-naught
- card(interval)= c
- Gordon Problems 1,4,7,8,11,18,19
(Assigned ; Due )
Chapter 1
- Section 1.5 Real-Valued Functions
- Definitions (Assume f:A->B)
- domain(f)
- range(f)
- home of the range of f
- f is increasing (strictly increasing) (decreasing) (strictly decreasing) (monotone)
- f is bounded (above) (below)
- f attains a max (min) at x=c.
- f attains a relative max (min) at x=c.
- f is a polynomial
- f is a rational function
- f is a trig function
- f is an exponential (log) function
- Theorems
- Gordon Problems 1,2,5,10,11,18,19,23,25
Chapter 2
- Section 2.1 Convergent Sequences
- Definitions
- Sequence
- {a_n} converges
- {a_n} diverges
- {a_n}->L
- {a_n}->+infinity (-infinity)
- a sequence is bounded (bounded above) (bounded below)
- a sequence is increasing (strictly increasing) (decreasing) (strictly decreasing)
- a sequence is monotone
- Theorems
- Limits are unique
- Convergent sequences are bounded
- Algebra of Convergent Sequences Theorem
- Two Trains Theorem (Squeeze Theorem) for Sequences
- a<=x_n<=b for every n and {x_n}->x, then a<=x<=b.
- Gordon Problems:
Set 1: 1,3,4,5,7,8,10,13;
(Assigned
; Due
)
Section 2.1: Set 2 15,16, 17,18,20,28,31,33
(Assigned
; Due
)
Chapter 2
- Section 2.3 ( Note the order change)
- Definitions
- subsequence
- limit inferior
- limit superior
- subsequential limit
- Theorems
- {x_n} converges if and only if every subsequence converges
- Every sequence has a monotone subsequence
- Bolzano-Weierstrass Theorem
- Gordon Problems: 1-4, 6,9,13,14,17,26,27
(Assigned
; Due
)
Chapter 2
- Section 2.2
- Definitions
- Cauchy sequence
- A nest of inervals (or a nested sequence of intervals)
- A potpouri of convergent sequences theorem
- Theorems
- If {a_n} is monotone, then {a_n} convergese <=> {a_n} is bounded.
- Cauchy sequences are bounded
- A sequence, {a_n} converges <=> {a_n} is Cauchy
- Nested Interval Theorem
- Gordon Problems: 1,3,4,5,9,10,14,16,18,24
Chapter 3
- Section 3.1
- Definitions
- limit of a function at a point, x_o (2 definitions)
- right and left hand limits of a function at a point, x_o
- Theorems
- Algebra of Limits Theorem for Functions (SAL)
- Squeeze Theorem for Functions
- Gordon Problems 2,6,9,10,13, 23,24,27,28,32
(Assigned
; Due
)
Chapter 3
- Section 3.2
- Definitions
- f is continuous at x_o (2 definitions)
- f:A->B is continuous
- f has a jump discontinuity at x_o
- f has a removable discontinuity at x_o
- Theorems
- Algebra of Continuous Functions Theorem (CAL)
- Polynomials are everywhere continuous
- Rational Functions are continuous except at poles.
- Gordon Problems 1,4,6,7,8,9,10,13,16,18
(Assigned
; Due
)
Chapter 3
- Section 3.3
- Definitions
- intermediate value of a function f:A->B
- f has the intermediate value property on a set A
- extreme value of a function f:A->B
- f is locally bounded
- the set E is compact
- Theorems
- Intermediate Value Theorem
- Extreme Value Theorem
- If the domain of a continuous function is compact, so is the range
- If the domain of a continuous function is an interval, so is the range
- If f is monotone, f has only jump discontinuities
- If f is monotone, f has at most countably many discontinuities
- A monotone function with the intermediate value property is continuous.
- A continuous strictly monotone function has a continuous strictly monotone inverse.
- Gordon Problems 1,3,5,6,7,11,13,16,18,27,34,39
(Assigned
; Due
)
Chapter 4
- Section 4.1
- Definitions
- difference quotient
- f is differentiable at x_o
- f is differentiable on and interval [a,b]
- Theorems
- Algebra of Differentiable Functions Theorem
- constant multiplier rule
- sum (or difference) rule
- product rule
- quotient rule
- humke rules
- chain rule
- Derivatives of Standard Functions Theorem
- Gordon Problems 1,2,4,6,10 (use MI),
(Assigned
; Due
)
12,16,2122,23,24
(Assigned
; Due
)
Chapter 4
- Section 4.2
- Definitions
- Theorems
- Rolle's Theorem
- Mean Value Theorem
- Monotonicity Theorem for Derivatives
- First Derivative Test
- Second Derivative Test
- L'Hosptal's Rule
- Gordon Problems 2,3,4,8,10,17,20,23
(Assigned
; Due
)
Chapter 7
- Section 7.1
- Definitions
- {f_{n}}->f pointwise on A
- {Sigma f_{n}}->f pointwise on A
- Gordon Problems 7.1A page 243 Examples 1-8 investigate and verify
(Assigned
; Due
)
- Gordon Problems 7.1B page 246 Problems 1-5,7,8,15
(Assigned
; Due
)
Chapter 7
- Section 7.2
- Definitions
- {f_{n}}->f uniformly on A
- {Sigma f_{n}}->f uniformly on A
- Theorems
- If {f_{n}}->f pointwise on an interval I and M_{n}=sup{|f_{n}(x)-
f(x)| : x in I}. Then {f_{n}}->f uniformly on I if and only if {M_{n}}->0.
- Dini's Theorem
- Gordon Problems 7.2 page 251 Problems 1,3,4,6,7,10
(Assigned
; Due
)
Chapter 5
- Section 5.1
- Definitions
- partition, norm of a partition, refinement, tagged partition, Riemann integrable function, Riemann integral, oscillation of a function on an interval
- Theorems
- Algebra of Riemann integrable functions theorem
- Gordon Problems Page 169 1,3,5,6,10, 16,17
(Assigned
; Due
)
- Section
- Definitions
- Theorems
- Gordon Problems
Disclaimer
Questions? Click here: humke@stolaf.edu