MATH 4130 “Honors Introduction to Analysis I” Section 2 Fall 2016 Malott Hall 205 Tuesday/Thursday 10:10–11:25 Professor: Robert Strichartz • Office: 563 Malott, Office Hours: Mon 10:30 – 11:30, Wed 10:30 – 11:30 If the weather is nice, office hours may be held in the flower garden between the AD White House and the Big Red Barn. • E-mail: [email protected]• Office Phone: 255-3509 T.A.: Avery St. Dizier • Office Hours: Mon 4:00 – 5:00, Wed 3:00 – 4:00 in Malott 218 • E-mail: [email protected]Text: R. Strichartz, The Way of Analysis, revised edition (2000) WARNING: DO NOT BUY THE FIRST EDITION. SOME PROBLEMS HAVE BEEN CHANGED. Interview: Please come for a short interview during the first week. There will be a sign-up sheet at the first class. Course description: Introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. Based entirely on proofs. The student is expected to know how to read and, to some extent, construct proofs before taking this course. Topics typically include construction of the real number system, properties of the real number system, continuous functions, differential and integral calculus of functions of one variable, sequences and series of functions. The course will cover the first eight chapters of the text. Grading and Exams: • Class participation (25%) • Written homework - weekly assignments due in class on Tuesdays (25%) • Midterm - in class October 6 (20%) • Final exam, to be scheduled (30%) ”A Cornell student’s submission of work for academic credit indicates that the work is the student’s own. All outside assistance should be acknowledged, and the student’s academic position truthfully reported at all times” ... Cornell University Code of Academic Integrity. 1
Transcript
MATH 4130 “Honors Introduction to Analysis I”
Section 2
Fall 2016
Malott Hall 205Tuesday/Thursday 10:10–11:25
Professor: Robert Strichartz
• Office: 563 Malott, Office Hours: Mon 10:30 – 11:30, Wed 10:30 – 11:30If the weather is nice, office hours may be held in the flower garden between the AD WhiteHouse and the Big Red Barn.
Text: R. Strichartz, The Way of Analysis, revised edition (2000)WARNING: DO NOT BUY THE FIRST EDITION. SOME PROBLEMS HAVE BEENCHANGED.
Interview: Please come for a short interview during the first week. There will be a sign-upsheet at the first class.
Course description:
Introduction to the rigorous theory underlying calculus, covering the real number system andfunctions of one variable. Based entirely on proofs. The student is expected to know how to readand, to some extent, construct proofs before taking this course. Topics typically includeconstruction of the real number system, properties of the real number system, continuousfunctions, differential and integral calculus of functions of one variable, sequences and series offunctions. The course will cover the first eight chapters of the text.
Grading and Exams:
• Class participation (25%)
• Written homework - weekly assignments due in class on Tuesdays (25%)
• Midterm - in class October 6 (20%)
• Final exam, to be scheduled (30%)
”A Cornell student’s submission of work for academic credit indicates that the work is thestudent’s own. All outside assistance should be acknowledged, and the student’s academicposition truthfully reported at all times” ... Cornell University Code of Academic Integrity.
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How the course will work:
For each class there will an assigned reading from the text and a set of discussion questions. Youwill do the reading and think about the questions before the class. Most of the class time will bedevoted to discussing the questions. We will also discuss questions that you bring up, and maygo over some homework problems. It is expected that everyone will participate in the discussion.Approximately 25% of your grade will be based on your enthusiastic participation.
The goal is to get each student to understand the material through a learning process that ismessy, challenging, individualized, frightening, thrilling and ultimately transcendental.
Homework:
You are allowed to work together with other students on HW, provided you write up thesolutions on your own. Please write the names of the students you worked with on the top ofyour HW paper.WARNING: numbers listed below are for the 2nd edition of the textbook. If youhave older edition, please, check if you are doing the right problems. A copy of the2nd edition of the textbook is available in the Malott library.
Problem 2.2.4(4): you can use infinite decimal expansions if you wish;Problem 2.2.4(9): there is a typo, parentheses need to be around x2 + y2;
HW 11
There is no class on December 6. However, HW 12 is due December 6.
Exams:
• Midterm Exam: In class October 6
• Final Exam: TBA
These will be closed book exams.
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Daily readings and questions:
August 25
Read 1.1, 1.2, 1.3, 1.4, 1.5
Discussion questions:
1. Does every mathematical statement have to have quantifiers (perhaps only implicitly)?
2. Use quantifiers explicitly to formulate the statement “there exist infinitely many primes.”Do the same for the negation of the statement.
3. Does the order within a sequence of quantifiers of the same type matter?
4. Does an infinite subset of an uncountable set have to be uncountable?
5. How many different ways can you “count” the set of integers?
6. In the proof of a theorem do you have to use all the hypotheses?
7. What does the Axiom of Archimedes say for the rational numbers? Can you prove it?
8. How many representations pq
are there for a fixed rational number? Is there a “best choice”?
9. Do you need the Axiom of Choice to make a finite number of choices?
10. To prove “a or b,” do you have to show explicitly which of the statements is true?
August 30
Read 2.1
Discussion questions:
1. What problem was Cauchy trying to solve with his criterion?
2. How do we know Q ⊂ R? Here Q denotes the rational numbers.
3. Given a Cauchy sequence of rationals, is there a way to tell whether or not it represents arational number?
4. In the proof of Lemma 2.1.1, we made the “tricky” choice of errors 12n
so that the ultimateerror was 1
2n+ 1
2n= 1
n. After the proof it is pointed out that we do not have to be “tricky.”
If the initial errors were chosen to be 1n, then the ultimate error is 2
nand this is just as
good. Why is it just as good? Which proof do you prefer?
5. What principle allows us to consider errors of the form 1n, 1m
instead of ε, δ? What do wegain by this?
6. Let xn = log n. Does limn→∞
xn exist? Does {log n} satisfy the Cauchy criterion? What is the
approximate size of log(n+ 1)− log n? Does it go to 0?
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7. What do we get if we consider the space of equivalence classes of Cauchy sequence ofintegers?
September 1
Read 2.2
Discussion questions:
1. If we were to replace the formula on the bottom of p.39, with
xjyj − xkyk = xj(yj − yk) + yk(xj − xk)
would it make any difference?
2. Are the proofs in this section merely “follow your nose,” or do they use surprising ideas?Which of them was the most difficult for you to understand?
3. Division by zero is forbidden in both Q and R. How does Lemma 2.2.4 allow us to definedivision by a nonzero real without getting hung up by division by zero in the Cauchysequences?
4. When we define an operation on real numbers, do you always have to check that thedefinition is independent of the choice of Cauchy sequence representing the number? Inpractice, is this ever difficult?
5. If x is represented by the Cauchy sequence {xn} and y by {yn}, what does the conditionxj < yj for all j (or all sufficiently large j) tell you about x and y?
6. True or false: in a Cauchy sequence, the first million terms don’t matter.
7. In what sense do the results of this section justify the definition of the real numbers inSection 2.1?
September 6
Read 2.3
Discussion questions:
1. Numbers of the form a+ b√
2, where a, b are rational, form a field that contains therationals and is contained in the reals. Could it be complete?
2. Why is it important to know that the reals are complete?
3. What is the intuitive idea behind the proof of completeness? Why can’t you modify theproof to show that the rationals are complete?
4. In the proof of Theorem 2.3.2, why is it OK if some of the quotients xk
ykare undefined?
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5. How does the “divide and conquer” method in the proof Theorem 2.3.3 produce a Cauchysequence of rationals converging to
√x? Why can’t you just take the Cauchy sequence
x1, x2, . . . defining x and replace it by√x1,√x2, . . . ?
6. Would you rather see an easier proof that leads to an inefficient algorithm, or a moredifficult proof leading to an efficient algorithm?
7. Do you think you can prove the existence of cube roots? More generally, n-th roots?
8. A dyadic rational number is a number of the form k2m
. What do you get if you consider allCauchy sequences of dyadic rational numbers?
September 8
Read 2.4
Discussion questions:
1. Is 0.19999 . . . the same as 0.200 . . . ?
2. Given an infinite decimal expansion of x, can you decide whether or not x is rational?
3. How many infinite decimals represent 0?
4. What do you like better, Dedekind cuts or equivalence of classes of Cauchy sequences ofrationals? Why?
5. What appeals to you more, the real number system, the nonstandard real number system,or the constructive real number system? Why?
6. How would you go from an infinite decimal expansion of x to the Dedekind cut Lx?Equivalently, given a Dedekind cut, could you use it to produce an infinite decimalexpansion?
September 13
Read 3.1
Discussion questions:
1. Why doesn’t +∞+ (−∞) make sense?
2. Why is inf E = − sup(−E)?
3. “The least upper bound of E is equal to supE.” Is this a theorem or a definition?
4. Are strict inequalities preserved in limits?
5. Why is the uniqueness part of Theorem 3.1.1 obvious?
6. Give an example of a sequence that has a limit but is neither monotone increasing normonotone decreasing.
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7. Why is it true that the sentence immediately following Definition 3.1.1 is equivalent to thedefinition?
8. What are the limit points of the sequence {1, 1, 2, 1, 2, 3, 1, 2, 3, 4, . . . } ?
9. In what sense is a lim sup a limit of sup’s?
10. Write yk = supj>k
xj. How is inf yk related to lim yk? You might first ask: what kind of
sequence is {yk}?
11. For a sequence {xj}, is it true that every value of t satisfying lim inf xj 6 t 6 lim supxjmust be a limit point of {xj}?
September 15
Read 3.2
Discussion questions:
1. What is the relationship between (0, 1) ∪ (1, 2) and (0, 2)?
2. Is the intersection of a finite number of open intervals equal to an open interval?
3. Why don’t we just define a closed set to be the complement of an open set?
4. Give an example of a set that is neither open nor closed.
5. Can you describe the complement of the Cantor set?
6. Is it true that the union of all open sets contained in A is equal to the interior of A? Howdoes this relate to the statement that the intersection of all closed sets containing A isequal to the closure of A?
7. Is it always true that A ∩Q is dense in A? (Remember that Q denotes the rationals.)
8. Is the empty set closed?
9. Why is a finite set always closed?
10. If you remove a finite set of points from an open set is the result still open? What aboutadding a finite set of points to a closed set?
11. Denote the complement of a set B by B′. What is (interior(A′))′? What is (closure(A′))′?
September 20
Read 3.3
Note that this is a very challenging section. There are essentially five proofs: (a), (b), (c)relatively easy
(a) compact ⇒ closed and bounded;
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(b) HB ⇒ closed and bounded;
(c) Theorem 3.3.3 (nested sequence property);
and (d),(e) relatively hard
(d) closed and bounded ⇒ compact;
(e) compact ⇒ HB
[Here HB means Heine–Borel property: every open covering has a finite subcovering.]
Please try to learn some of the proofs well enough to present them at the blackboard. I will askfor volunteers to do this.
Discussion questions:
1. Is the empty set compact?
2. In Definition 3.3.1, why is it required that the limit points belong to A? What wouldhappen if we drop this requirement?
3. The first part of the proof of Theorem 3.3.2 reduces to the case of countable open covers.Does this argument uses the compactness of A?
4. Before the proof of Theorem 3.3.2 it is shown that HB ⇒ closed and bounded. It is alsopossible to show that HB ⇒ compact. Can you fill the details in the following argument:
Suppose A satisfies HB, and suppose x1, x2, . . . is a sequence of points in A with no limitpoint in A. We will show that this is impossible. For each a ∈ A, there exists 1
n(depending
on a) such that the interval (a− 1n, a+ 1
n) contains only a finite number of x1, x2, . . . . Call
Ia the interval (a− 1n, a+ 1
n). Then the sets Ia (as a varies in A) form an open cover of A.
By HB there is a finite subcover, say Ia1 , . . . , Iam , so A ⊆ Ia1 ∩ · · · ∪ Iam . How many of thex1, x2, . . . can belong to A?
September 22
Read 4.1
Discussion questions:
1. What can you say about the intersection of the graph of a function with a vertical line? ahorizontal line? Does it matter whether or not the function is continuous?
2. Is an implicit description of a function unique? (i.e. can there be more than one?)
3. Is the composition of continuous functions necessarily continuous?
4. What is “uniform” about uniform continuity?
5. Is f(x) = x2 with domain [−1, 1] uniformly continuous? What if the domain is R?
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6. Informally, continuity means that you can control the error in the output ( f(x) ) bycontrolling the error in the input ( x ). Can you use this informal description to distinguishbetween continuity at a point, continuity on the whole domain, and uniform continuity?
7. Explain why, for continuous functions, it is the image of a convergent sequence thatconverges, while it is the inverse image of an open set that is open.
8. Give an informal description of the Lipschitz condition in terms of “stretching.”
9. Why do we want to exclude the value f(x0) in the definition of limx→x0
f(x)?
September 27
Read 4.2
Discussion questions:
1. How do you express min(f, g) in terms of max and minus signs?
2. There is an easier proof of Theorem 4.2.1. Consider separately the cases f(x0) = g(x0) andf(x0) > g(x0). Try to give the argument for each case. In the second case, ask yourself iff(x) > g(x) for x near x0.
3. Why do you think the function shown in Fig.4.4.2 is Lipschitz? What Lipschitz constantwould work?
4. Does the proof of the Intermediate Value Theorem give an effective way to find x?
5. Do the proofs of Theorems 4.2.3, 4.2.4 and 4.2.5 use the closed and bounded equivalentcondition for compactness?
6. If f is continuous, must the inverse image of a compact set under f be compact?
7. Combine Theorem 4.2.3 and the Intermediate Value Theorem to show that a continuousfunction maps a closed bounded interval to a closed bounded interval.
8. For a monotone function, does f(x0) have to lie between the two 1-sided limits?
9. How does the last line of the proof of Cor. 4.2.1 yield the conclusion?
September 29
Read 5.1
Discussion questions:
1. In Definition 5.1.1 on the top line of p.145 it says x 6= x0. Nevertheless f(x0) must bedefined in order for f ′(x0) to be defined. Is there a contradiction here?
2. Of the two formulas on the top of p.145, which one do you like better? Why?
3. Could you define one-sided derivatives, analogous to one-sided limits?
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4. Can a tangent line be vertical? Can a vertical line be tangent to the graph of a function?Are these the same or different questions?
5. Can you express the Lipschitz condition in terms of “big Oh” and “little oh” notation?
6. Does the formula on the bottom of p.150 make sense for x = 0? Why, or why not?
7. Could you define a tangent line as a line that stays on one side of a graph?
8. Is our definition of f ′(x0) any different from the definition in a standard calculus book?
9. If you create a function h(x) by gluing together differentiable functions
h(x) =
{f(x), a 6 x 6 b
g(x), b 6 x 6 c
what conditions do you need for h to be differentiable at b?
October 4
Read 5.2
Discussion questions:
1. Is the condition “f is monotone increasing at every point in the interval (a, b)” the same as“f is monotone increasing on the interval (a, b)”? What about if you replace “monotoneincreasing” by “strictly increasing” in both statements?
2. Does monotone or strictly increasing imply continuity?
3. Is there a “proof theme” in this section: “do 0 first”?
4. Is Fermat’s method genuinely different from solving f ′(x) = 0?
5. Why is Theorem 5.2.2 a “global” result, while Theorem 5.2.1 is a “local” result ?
6. How does Theorem 5.2.2(a) give us a stronger result than anything in Theorem 5.2.1?
7. Does the MVT require one-sided derivatives to exist at the endpoints?
8. If the hypotheses of MVT hold on an interval [a, b], does it follow that they hold on anysubinterval [c, d] ⊂ [a, b]? What would the conclusion of MVT on all subintervals say?
9. Is the point x0 whose existence is asserted in the conclusion of the MVT necessarily unique?
October 6In class prelim. Closed book. Covers materials through section 5.2.
October 13
Read 5.3
Discussion questions:
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1. In the proof of Theorem 5.3.1 for quotients, we use the fact that differentiability impliescontinuity. Do we need to use this fact to prove the result for sums? For products?
2. In the chain rule, why are f ′ and g′ are evaluated at different points?
3. Give an intuitive argument for the chain rule based on the intuitive idea that the derivativeof a function is a multiplicative factor for differences.
4. Is the composition of two Lipschitz functions necessarily Lipschitz? What can you sayabout the Lipschitz constants?
5. What is the relationship between the two identities f−1 ◦ f = ID and f ◦ f−1 = IR?
6. If you know that the Local Inverse Function Theorem was true, could you use it to easilyobtain the Inverse Function Theorem?
7. What is the relationship between the chain rule for f−1 (f(x)) = x and the InverseFunction theorem?
8. Why is switching the x-axis and y-axis equivalent to reflecting in the line y = x?
9. Can f−1 exist if f ′(x) = 0 for some points x?
October 18
Read 5.4
Discussion questions:
1. Is Theorem 5.4.1 parts (c) and (d) the same as the second derivative test in calculus?
2. What is limh→0
f(x+h)−2f(x)+f(x−h)h2 ?
3. Why is o(|x− x0|2) a stronger condition than o(|x− x0|) but a weaker condition thano(|x− x0|3)?
4. What does Taylor’s theorem tell you if f is C4, f ′(x0) = f ′′(x0) = f ′′′(x0) = 0 andf (4)(x0) > 0?
5. There is a typo in the middle of p.187 in the formula for f(x)g(x)
. The middle term is incorrect.What should it be?
6. Let f(x) =
{sinxx, if x 6= 0
1, if x = 0. What is f ′(0)?
7. The Lagrange Remainder Formula is an example of the cliche “assume more, get more.”What more are you assuming? What more are you getting?
8. If f is Ck+1, what is the relationship between Tk+1(x0, x) and Tk(x0, x)?
9. If f is Ck and x0, x1 are distinct points, what is the relationship between Tk(x0, x) andTk(x1, x).
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October 20
Read 6.1
Discussion questions:
1. In defining the integral, do the subintervals have to have equal lengths? Do the lengthshave to go to zero?
2. Why do we insist on considering all partitions with maximum length going to zero? Wouldit suffice to know the limit exists for one particular sequence of partitions?
3. How does allowing arbitrary evaluation points in subintervals pay off in the proof of theIntegration of the Derivative Theorem?
4. Why is there a factor of 1(n+1)!
in the Lagrange Remainder Formula and a factor of 1n!
in theIntegral Remainder Formula?
5. Think of the integration by parts formula as giving some information about g′ in terms ofexpressions involving only g, for different choices of f . What kind of information is this?
6. What does the change of variable formula say for f ≡ 1?
7. In the change of variable formula, what do the Cauchy sums for each integral look like?Can you relate them?
October 25
Read 6.2, 6.3
Comments on Lemma 6.2.1Let Ik = [xk−1, xk] and I ′j = [x′j−1, x
′j]. Say Ik meets I ′j if Ik ∩ I ′j is nonempty. Because the lengths
of the I ′j are less than the lengths of the Ik, there are at most two Ik that meet any given I ′j.Either I ′j ⊆ Ik for some k or I ′j ⊆ Ik ∪ Ik+1.
Why is M ′j −m′j ≤ (Mk −mk) + (Mk+1 −mk+1) in the second case?
xk−1 x′j−1 xk x′j xk+1
Because xk is in both Ik and Ik+1, so mk ≤ f(xk) ≤Mk and mk+1 ≤ f(xk) ≤Mk+1. Theintervals [mk,Mk] and [mk+1,Mk+1] overlap and the interval [m′j,M
′j] is contained in their union.
mk Mk
m′j M ′
j
mk+1 Mk+1
Of course in the first case I ′j ⊆ Ik, so we have M ′j −m′j ≤Mk −mk because we are taking sup
and inf over a larger interval. In either case,
M ′j −m′j ≤
∑Ik meets I′j
Mk −mk (one or two terms in the sum).
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Then,
Osc(f, P ′) =∑j
(M ′j −m′j)(x′j − x′j−1)
≤∑j
∑Ik meets I′j
(Mk −mk)(x′j − x′j−1)
Since these are finite sums, we can change the order of summation to get
Osc(f, P ′) ≤∑k
(Mk −mk)∑
I′j meets Ik
(x′j − x′j−1).
( )( )( )( ( ))xk−1 xk
The intervals I ′j that meet Ik must be smaller in length. They cover Ik and may overlap at eachend. The sum of the lengths of the I ′j meeting Ik is equal to xk − xk−1 plus the lengths of theoverlaps, which are each less than xk − xk−1. Thus∑
I′j meets Ik
(x′j − x′j−1) ≤ 3(xk − xk−1),
which gives
Osc(f, P ′) ≤ 3∑k
(Mk −mk)(xk − xk−1) ≤ Osc(f, P ).
Discussion questions:
1. There is a factor of 3 in Lemma 6.2.1. Where does it come from? Why doesn’t it matter?
2. In Theorem 6.2.1 part (c), what is the relationship between the values of the inf = sup andthe integral?
3. If you attempted to graph Dirichlet’s function, what would the graph look like?
4. What is the analogy between Theorem 6.2.1 parts (a) and (b) and the Cauchy criterion?
5. The integral of a product exists, but there is no formula for it. How would you explain thisto a calculus class? Is the same true for |f | and max(f, g) in Theorem 6.2.2?
6. There is a typo in line 5 page 229. What is it?
7. Do the numerical integration formulas in Section 6.1.4 work for the Riemann integral?
8. If you change the values of a function at a finite number of points, does that change theintegral? What about changing values at a countable number of points?
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9. For which values of a does the improper integral∞∫0
xa
1+x2dx exist?
October 27
Read 7.1
Discussion questions:
1. The complex number system is complete in two different senses. What are they, and howare they different?
2. Can you give a geometric proof of the triangle inequality?
3. What are all possible values of |z1 + z2| if |z1| = r1 and |z2| = r2?
4. Give a geometric interpretation of the transformation z 7→ iz.
5. How would you relate C and the two-dimensional real vector space R2?
6. How can you tell the difference between the complex numbers i and −i?
7. The Mean Value Theorem does not hold for complex valued functions. What about theintegration of the derivative theorem? Is there a moral to this story?
8. Do you think that |F | integrable implies F integrable?
9. In what way is the inequality in Theorem 7.1.1 related to the triangle inequality?
November 1
Read 7.2
Discussion questions:
1. What is the sum of the geometric series∞∑k=0
rk if we start the sum at k = 0 rather than
k = 1?
2. For an absolutely convergent series, is there any relationship between∞∑n=1
xn and∞∑n=1
|xn|?
3. If you want to use the comparison test to prove convergence of a given series∞∑n=1
xn, is
there any strategy to pick the values of yn?
4. On p.255, why is it sufficient to understand the behavior of the partial sumsk∑
n=1
bn for
k = 2m − 1?
5. To study the convergence of∞∑n=1
1na we did not use comparison with a geometric series
directly, but in the end we did end up using it. Explain.
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6. Absolute convergence is equivalent to unconditional convergence. Which property do youthink is more useful? Which property do you think is easier to verify?
7. Why is the estimate at the bottom of p. 257 true?
8. In Theorem 7.2.5, is part a. a special case of part b.?
9. The integration by parts formula has boundary terms. What are the analogous terms inthe summation by parts formula?
November 3
Read 7.3
Corrections for p.273 and 274:
Conversely, assume fn(xn)→ f(x) if xn → x. First, we claim that f is continuous. That meansthat we have to show f(xn)→ f(x), or equivalently, f(xn)− f(x)→ 0. But look at the doublecomparison f(xn)− f(x) = [f(xn)− fm(xn)] + [fm(xn)− f(x)] where m depends on n as follows.Given an error 1
kwe have |f(xn)− fm(xn)| ≤ 1
2kfor all sufficiently large m, because fm → f at
xn. Choose one such m, call it m(n). We need to show that |fm(n)− f(x)| ≤ 12k
for all sufficientlylarge n. Define a new sequence of points yj as follows:
yj =
{xn if j = m(n)
x if j 6= m(n) for any n
(we can easily arrange that all m(n) are distinct). Clearly yj → x so fj(yj)→ f(x). But{fm(n)(xn)} is a subsequence of {fj(yj)} (corresponding to j = m(n)). So |fm(n) − f(x)| ≤ 1
2kfor
all sufficiently large m. Thus f is continuous. [A curious observation is that this part of theargument did not use the continuity of the functions fn.]To show fn converges uniformly to f we consider the sets
Am,N = {x ∈ D such that |fn(x)− f(x)| ≤ 1
m}
The condition for uniform convergence that we want to prove is that for all m there exists Nsuch that Am,N = D. Ordinary pointwise convergence would say that for all x and all m thereexist N such that x is in Am,N . But we are assuming more than pointwise convergence, and weclaim the following stronger conclusion: for each x in D and every m, there exists N such thatAm,N contains a neighborhood of x.We prove this claim by contradiction. If it were not true, then there would exist a fixed x and mand a sequence of points {xN} converging to x such that xN is not in Am,N . But xn not in Am,N
means there exists k(N) ≥ N such that |fk(N)(xN)− f(xN)| > 1/m. Since f is continuous we canmake |f(xN)− f(x)| ≤ 1/2m for N large, so |fk(N)(xN)− f(x)| > 1/2m by the triangle inequality.By inserting some terms equal to x into the sequence {xn} we can obtain a new sequence {yk}converging to x such that |fk(yk)− f(x)| > 1/2m for infinitely many k. Specifically, we takeyk = xN if k = k(N), choosing the smallest N if there is more than one choice and yk = xotherwise. But this contradicts the hypothesis lim
k→∞fk(yk) = f(x), proving the claim.
Discussion questions:
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1. In Theorem 7.3.1, it is observed that the Cauchy criterion for the sequence of functionsimplies the Cauchy criterion for the sequence of values at any given point x. Is it true thatthe Cauchy criterion for {fn(x)} for every x implies the Cauchy criterion for the sequenceof functions?
2. Suppose fn → 0 uniformly. What can you say about the graphs of the functions fn (interms of containment in rectangles)?
3. Can a uniform limit of discontinuous functions be continuous?
4. In Theorem 7.3.3, could we conclude that limn→∞
b∫a
|fn(x)− f(x)|dx = 0?
5. Which part of the Fundamental Theorem of Calculus is used in the proof of Theorem 7.3.4?
6. Suppose the functions fn are assumed to be in C2 [remember this means twicecontinuously differentiable], with fn → f pointwise. What would we need to assume inorder to conclude that f is C2?
7. In the proof of Theorem 7.3.5 in the top of p.273 there are two displayed inequalities. Whyis the second one useful while the first one is not? Is the first inequality still true?
8. In the proof of the converse portion of Theorem 7.3.5, what are we assuming that isstronger than pointwise convergence? In particular, why does it imply pointwiseconvergence?
November 8
Read 7.4.1, 7.4.2
Discussion questions:
1. Can you interpret the geometric series as a special case of power series?
2. Does the radius of convergence of a power series depend on the first N terms?
3. Suppose you multiply a power series by a polynomial. What do you get?
4. Suppose you add two power series about a point x0. Is the result a power series, and whatcan you say about the radius of convergence?
5. Compare the properties of power series and Taylor expansions. In what ways are theysimilar? In what ways are they different?
6. The formula for bk pn p.282 only depends on an for n ≥ k. Why?
7. Is the formula on p.284 for the power series of 11−x about x0 meaningful for x0 > 1?
8. What is the power series for 11+x2 about x = 0? What is its radius of convergence?
9. Where is f(x) = |x| analytic?
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November 10
Read 7.4.3, 7.4.4, 7.5.1
Discussion questions:
1. Using the “distance to the nearest complex singularity” method, what should the radius ofconvergence of 1
1+x2 about x0 be?
2. Why is the formula on the top of p.288 true? Why is it < (1 + x20)n?
3. What should the power series of 11+x2 about x0 be, using the derivatives formula?
4. What should the radius of convergence of f(x) = 1p(x)
about x = x0 be, where p(x) is apolynomial?
5. Why is the power series for 1g(x)
a special case of Theorem 7.4.5?
6. Can you use the chain rule to determine the power series of g ◦ f?
7. Is the polynomial of degree n− 1 that interpolates n data points unique?
8. qk and Qk are polynomials with only real roots. Is the same true for the Lagrangeinterpolation polynomial P?
November 15
Read 7.5.2, 7.5.3, 7.5.4 and the revised version of p.300 (see below). Note: the results on p.300in the book are correct, but the definition of approximate identity given is too restrictive, and isnot strong enough for the application in 7.5.3.
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The following is the correct version of page 300.All the corrections are colored in red.
because I have been vague as to what is meant by the condition that gngets more concentrated near zero as n→∞. In fact, depending uponthe context, there are several ways of making this precise. Here is one:
suppose gm ≥ 0 satisfies∫gm(y)dy = 1 and
∫ −1/n−∞ +
∫∞1/n
gm(y)dy → 0
as m→∞ for all n. Then
|f ∗ gm − f(x)| =∣∣∣∣∫ ∞−∞
(f(x− y)− f(x))gm(y)dy
∣∣∣∣≤∫ −1/n−∞
+
∫ ∞1/n
|f(x− y)− f(x)|gm(y)dy
+
∫ 1/n
−1/n|f(x− y)− f(x)|gm(y)dy .
We are assuming f is continuous and vanishes outside a bounded inter-val. This implies that f is bounded and uniformly continuous. Thus|f(x)| ≤M for all x, and given any error 1/k there exists 1/n such that|f(x− y)− f(x)| < 1/k for all |y| ≤ 1/n. Substituting these estimatesinto the integrals (using |f(x− y)− f(x)| ≤ |f(x− y)|+ |f(x)| ≤ 2Min the |y| > 1/n integral) we obtain
|f ∗ gm(x)− f(x)| ≤ 2M
∫|y|≥1/n
gm(y)dy +1
k
∫ 1/n
−1/ngm(y)dy
≤ 2M
∫|y|≥1/n
gm(y)dy +1
k
since∫ 1/n
−1/n gm(y)dy ≤∫∞−∞ gm(y)dy = 1. But we are assuming∫
|y|≥1/n gm(y)dy → 0 as m→∞ (this is our concentrating hypoth-
esis), so we can make 2M∫|y|≥1/n gm(y)dy ≤ 1/k by taking m large
enough and, hence, |f ∗ g(x)− f(x)| ≤ 2/k if m is large enough. Wecan summarize this result as follows:Definition 7.5.1 A sequence of continuous functions on the line {gm}satisfying
1. gm(x) ≥ 0,
2.∫∞−∞ gm(x)dx = 1,
3. limm→∞
∫|x|≥1/n gm(x)dx = 0 for all n
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Discussion questions:
1. In the first displayed equation on p.299, how is the “fair” averaging property used?
2. In the corrected p. 300 there are two parameters named n and m. Which one goes toinfinity first?
3. If f is increasing and g is one of the gm from an approximate identity, is f ∗ g increasing?
4. Can you give an intuitive explanation why f ∗ g is a polynomial if g is a polynomial?
5. What is the intuitive idea behind the proof of the approximate identity lemma?
6. Is the proof of the Weierstrass Approximation Theorem constructive?
7. In the displayed equation at the bottom of p. 304, what does the constant c depend on?
8. If f is C1 and you want to approximate both f and f ′, can you use the same polynomialsas in the Weierstrass Approximation Theorem?
9. If f is Ck and g is Cm, how smooth is f ∗ g?
10. If f is defined and continuous on [a, b] ∪ [c, d] for b < c, can a single sequence ofpolynomials approximate f on both intervals?
November 17
Read 7.6
Discussion questions:
1. How would you formulate a notion of equicontinuity for a sequence of functions fn at asingle point x0?
2. In the concept of uniform equicontinuity, what does the “uniform” refer to? What does the“equi-” refer to?
3. For a sequence {xn} of real numbers, what condition implies the existence of a convergentsubsequence?
4. In the infinite matrix on p.312, how are the rows related to each other? What about thecolumns?
5. Where in the proof of the Arzela–Ascoli Theorem is the uniform equicontinuity hypothesisused?
6. We say that a sequence of functions {fn} is equi-Lipschitz if there exists a constant M suchthat |fn(x)− fn(y)| ≤M |x− y| for all x, y and n. How does this concept relate to uniformequiconinuity? How does it relate to the uniform boundedness of the derivatives of {fn}?
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November 22
Read 8.1
Discussion questions:
1. There are at least two formulas for e. what are they?
2. Can you deduce ex > 0 for x < 0 from the power series?
3. What are all possible solutions to the differential equation f ′ = f?
4. We have a power series for ex about x = 0 and log(1 + x) about x = 0. What do you get ifyou substitute both power series into log(1 + (ex − 1))?
5. If we were to take formula in Theorem 8.1.5 as the definition of log x, what would that tellus about d
dx(log x)? About the derivative of the inverse function?
6. Why should you expectN∑k=1
1k≈ logN?
7. Compare regularization with the Weierstrass approximation theorem.
8. If f and g are C∞ functions with the same Taylor polynomials about x0 for all orders, doesthis imply f = g in a neighborhood of x0?
9. In what sense is the infinite series defining f at the beginning of the proof of the BorelTheorem 8.1.7 really a finite sum?
November 29
Read 8.2
Discussion questions:
1. Would the formula π = 2 arcsin 1 be a reasonable way to compute π using numericalintegration?
2. The function arcsinx is not differentiable at x = 1. Can you relate this to properties ofsinx via the inverse function theorem?
3. Can you deduce C(x)2 + S(x)2 = 1 from the power series on p.343?
4. What are all solutions to the differential equation f ′′ = −f?
5. Why should we believe that exp(z1 + z2) = exp(z1) exp(z2) for complex numbers z1, z2?
6. What is∫∞0
11+t2
dt?
7. Why is tan θ periodic of period π while sin θ and cos θ are periodic of period 2π?
8. If you were to graph the partial sums of the power series for C(x) and S(x), would you beled to conjecture that C(x+ 2π) = C(x) and S(x+ 2π) = S(x)?
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9. What differential equation does f(x) = tan x satisfy?
