#### Spring-2015. Introduction to Mathematical Analysis (ccn: 54098)

Instructor: Alexander Givental
Lectures: TuTh 12:30-2, 289 Cory Hall
Office hours: Wed 1-3, in 701 Evans; on Wed morning by appointment.
GSI for Math 104 this semester is Shiyu Li who holds office hours on TuTh 9am-2pm in 1042 Evans.
Textbook (optional): W. Rudin, Principles of Mathematical Analysis

This course will be oranized in an experimental fashion, which clearly separates different forms of work from each other, as well as passing requirements from letter grade criteria. More precisely, there will be three components of this course: (a) theory, (b) technical skills, and (c) more conceptual problem solving.
(a) To pass this course, you will have to complete the theoretical part, namely, to write your own self-contained exposition of one-varible Mathematical Analysis. You may think of it as a required "term paper", but let us call it Lecture Notes . The major portion of in-class time and homework activity will be concentrated around this goal.
(b) In addition, there will be three homework assignments in the form of problem sets focusing on specific computational techniques; each assignment will be then folowed by a 10-15-min P/NP quiz, and a make-up quiz, if needed; each "pass" will add 1/3 of a letter grade point
(c) The rest of your letter grade (beyond the passing requirement and the quizes) will be based on the 3-hour written final exam (organized the usual way), where your problem-solving skills will be tested; it will be worth 5/3 of a letter grade point.
For instance, by completing the theory part, and passing all 3 quizzes, you'd earn at leat a B- for the course even before the final, but failing all 3 quizzes would limit your course grade by a B+ at best.
In addition to the theory of one-variable calculus, you are encouraged to study more advanced topics from our textbook (such as e.g. the topology of metric spaces) in the regime of a reading course; we don't plan however to factor this work into passing or letter grade criteria.

Tentative weekly plan:

Week

Jan 20 22: Rational and real numbers / Countable and uncountable sets
Jan 27 29: Open and closed sets / Heine-Borel's thm. Compactness. Weirestrass' theorem
Feb 3 5: Limits of sequences / Subseqences, lower and upper limits
Feb 10 12: Cauchy sequences, completeness. Connectedness / Special sequences
Feb 17 19: Limits of functions / Continuity
Feb 24 26: Uniform continuity, maxima and minima / Derivatives
Mar 3 5: Ferma's, Rolle's, Lagrange's, and Taylor's thms / l'Hospital's rule. Computation of limits using Taylor's formula
Mar 10 12: Riemann integral / Properties of integrals
Mar 17 19: The fundamental thm of calculus / Techniques of integration
[Mar 24 26: Spring break]
Apr 31 2: Infinite series; Cauchy, root, ratio, integral, etc. tests
Apr 7 9: Power series
Apr 14 16: Sequences and series of functions. Uniform convergence
Apr 21 23: Equicontinuity, Arzela-Ascoli's thm / Weierstrass' approximation thm
Apr 28 30: Review / Exercises
May 5 7: (RRR week) Exercises
May 14, Th, 3-6 pm: Final exam

#### HOMEWORK

From Lecture 1: Begin typesetting your personal exposition of the theory: the "Lecture Notes." The recommendation is to install LaTeX on your computer, for instance, following the instructions at the Art of Problem Solving website.
The introductory section of your notes should contain
• an example of a polynomial y=f(x) with rational coefficients which for rational values of x from the interval [0,1] does not achieve the maximum or minimum value;
• statement and proof of the theorem that fractions are represented by repeating decimals, and vice versa , i.e. that a repeating decimal represents a fraction;
• definition of real numbers as signed decimal sequences infinite to the right (with the approriate identification between tails of 9s and 0s).
From Lecture 2: I think, your Lecture Notes should include the following results (formulations and proofs):
• the union of at most countably many of at most countable sets is countable at most (and the corollary that Q is countable)
• the unit square on the real plane has the same cardinality as an interval on the real line (called continuum)
• continuum is uncountable.
This roughly corresponds to the material on pp. 24-30 of the textbook
HW1: You are expected to email me the first portion of your notes by Friday night. It should include the matherial of Lecture 1 (or both 1 and 2). I've aleady started getting some samples, and would like to make here some comments (which probably would apply to many) so that I dont have to repeat them to each one of you individually.
In principle each portion of a mathematical text can be nothing but: Definition, Theorem (i.e the formulation of it), or Proof. For expositional purposes one might also add: Remark, Example (which in fact is also a definition, theorem, and proof together), Exercise (i.e. the formulation of it). So, structure your text accordingly. If a sentense you are writing doesn't fit any of these types, it better be removed.
Remark on LaTeX: Chris Dock sent me the following link where you can (if you don't want to install LaTeX on your computer) remotely typeset a text rendering portions of it in LaTeX: https://www.overleaf.com/

From Lecture 3: Include in your notes
• Definitions of neighborhoods, limit points, open sets, and closed sets in R;
• Proposition: unions and finite intersections of open sets are open;
• Proposition: complements of open sets are closed, and vice versa ;
• Corollary: intersections and finite unions of closed sets are closed;
• Definition of the closure of a set as the intersecion of all closed sets containig it, and the proof that this intersection is obtained from the set by including all its limit points;
• Example of the Cantor set, and the poof that it is closed, and has the cardinality of continuum (in our text, this example is discussed on p. 41, while the general material on open and closed sets on pp. 32-35)

From Lecture 4: Include in your notes (I am using LaTeX notations):
• Theorem: A non-empty subset $A \subset {\mathbb R}$ bounded above has the least upper bound (denoted $\sup A$)
• Corollary: A nested sequence of closed intervals has non-empty interection
• Definition: A set $A \subset {\mathbb R}$ is compact if every open cover of it has a finite subcover.
• Heine-Borel's Thorem: A closed interval [a,b] is compact
• Corollary: $A\subset {\mathbf R}$ is compact if and only if it is closed and bounded
• Corollary (Weierstrass' Theorem): An infinite bounded set has a limit point
Exercises useful for this theme, and better be attempted (and even better solved) by the next Lecture
• Given an unbounded subset in ${\mathbb R}$, give example of an open cover which does not have a finite subcover;
• Given a non-closed set, give example of an open cover which does not have a finite subcover;
• Prove that a sequence of non-empty nested compact sets has non-empty intersection;
• Generalize the theory to describe all compact subsets of the plane.
HW2 (due by Friday night, Jan. 30): write your notes at least up to Lecture 3 inclusively.

From Lecture 5: Finish (and correct) your exposition of the previous material; add definition of limit of sequences of real numbers, and prove properties of limits under the operations of addition, multiplication, and division of sequences (pp. 49-50 in Rudin's book).

From Lecture 6: Include: definition of a subsequence; properties:
• every subseququence of a convergent sequence converges to the same limit
• the set of subsequential limits of a given sequnce is closed
• (a variant of Weierstrass' theorem) a bounded sequence has a convergent subsequence
and definition of upper / lower limit (lim sup / lim inf) of a bounded sequence.

From Lecture 7: Definition of Cauchy secquences. Propositions: A converging sequence is Cauchy; a Cauchy sequence which has a convergent subsequence, converges.
Definition of a complete set (in R). Theorem: A subset in \R is complete if and only if it is closed. Corollary: Every Cauchy sequence in \R converges.
Definition of a connected set (in R). Theorem: A subset in \R is connected if and only if together with every two points x,y it contains the entire interval [x,y] (see section "Connectedness" in Rudin's book).
Exercises (for you to solve). Prove:
• The limit of a converging sequence is unique
• An increasing sequnce converges if an only if it is bounded above
• (two policemen's principle): If a_n < b_n < c_n for all n, while sequences a_n and c_n converge to the same limit, then b_n converges to the same limit
• X is closed if and only if the limit of every convergent sequence in X lies in X

From Lecture 8: Special sequences (see Rudin's book).

From Lecture 9: Limits of functions.
For a function f: X --> |R, x_0 a limit point of X \subset |R, and a number A, define what it means that lim_{x --> x_0} f(x) =A.
Proposition: This is equivalent to saying that for every sequence (x_n) in X\{x_0} which tends to x_0, the sequence (f(x_n)) tends to A.
Corollary about limits of sums, products, and ratios of two functions which are known to have limits as x tends to x_0.
Definition of continuity of a function f: X --> R at x_0 \in X.
Proposition': f is continuous at x_0 if and only if lim f(x_n) = f(lim x_n) for every sequence (x_n) in X converging to x_0.

From Lecture 9: Continuity.
Equivalence of four definitions of continuous functions X --> |R (in terms of \epsilon-\delta, limits of sequences, closed sets, open sets).
Corollary: Continuity of sums, products, and ratios of continuous fnctions.
Cotinuity of the composition of continuous functions.
Examples: Continuity of constant, linear, polynomial, rational functions; continuity of arithmetic operations.

From Lecture 10:
Continuity and compactness.
Theorem. A continuous image of a compact set is compact.
Corollary. A continuous real-valued function on a compact set achieve its maximum and minimum values.
Corollary. A continuous fucntion f: [a,b] --> |R achieves its maximum and minimum values.
Theorem. A continuous function on a compact set is uniformly continuous.
Continuity and connectedness.
Theorem. A continuous image of a connected set is connected.
Corollary. A continuous real-valued function on a connected set between any two values assumes all intermediate ones.
Corollary. The range of a continuous function f: [a,b] --> |R is the closed interval [m,M] between the minimum and maximum values.
Corollary. A continuous f: [a,b] --> |R has a zero on (a,b) if f(a) and f(b) have opposite signs.

From Lecture 11: Differentiability.
Definition of the derivative at a point.
Differentiability at a point implies continuity at this point.
Formulas for the derivatives of the sum, product, ratio, and composition of two differentialble functions.
Example of a function |R --> |R differentiable everywhere, whose derivative is discontinuous at the origin.

From Lecture 12:
Definition of global and local maximum and minimum.
Fermat's, Rolle's, Lagrange's and Taylor's theorems.
Exercise: Apply Taylor's formula to f(x)=(1+x)^a up to order x^n and prove, using an estimate for the remainder of order x^{n+1} that as n tends to infinity, the Taylor series converges to f(x) provided that |x|<1.

From Lecture 13:
Generalized Mean Value Theorem, l'Hospital's Rule, Taylor's formula with remainder in the form of Cauchy, an example of application to computation of limits.
Here is the practice set on limits of functions , and the answers are:
a/b, ln 2, m/n, mn(n-m)/2, n(n+1)/2, 1/2, 1, 1/4, 1, 17/5
Quiz on this matrial is scheduled for Th, April 2.

From Lecture 14: Integration (of bounded function f:[a,b] -> [m,M])
Notions of partition of [a,b] , lower and upper Riemann sum L(P,f), U(P,f), lower and upper integral as sup_P L(P,f) and inf_P U(P,f) respectively. Riemann integrability (as equality between lower and upper integrals).
Proposition about increase of L(P,f) and decrease of U(P,f) under refinement of partitions, the inequality L(P_1, f)\leq U(P_2,f) for arbitrary partitions P_1, P_2, and (in cosequence) the respective inequlity between the lower and upper integrals.
The Riemann integrability criterion: Existence for every \epsilon>0 of a patition P such that U(P,f)-L(P,f) < \epsilon.
Theorems about Riemann intgrability of functions: (a) continuous on [a,b], (b) bounded functions having at most finitely many discontinuity points, (c) monotone on [a,b].
Remark. Our exposition essentially follows Rudin's book, which however, gives a more general construction of "Riemann-Stieltjes integral" \int_a^b f(x) d\alpha, where \alpha is an increasing function on [a,b]. The theory of Riemann integral \int_a^b f(x) dx is extracted from Rudin's description by seting \alpha (x) = x. Beware though: Streamlining Rudin's exposition in this special case can be tricky.

From Lecture 15: Properties of integrals
Let f, f_1,f_2 be Riemann-integrable on [a,b], then
f_1+f_2 and cf are (and the repective equalities for the integrals hold true);
if f_1\leq f_2, then \int f_1 dx \leq \int f_2 dx
if a < c < b, then f is Riemann-integrable on [a,c], [c,b], and the integral is additive with respect to the interval of integration.
Theorem about Riemann-integrability of the composition of continuous function with a Riemann-integrable one, and the corollaries about integrability of the product of two Rimann-integrable functions, and of the absolute value of a Riemann-integrable function.

From Lecture 16: Integration and differentiation.
Change of variables in Riemann integrables.
The Fundamental theorem of calculus (if F'=f \in R[a,b], then \int_a^b fdx = F(b)-F(a)).
Another form: if f \in R[a,b], then F(x):=\int_a^x fdx is continuous, and d F/dx (x) = f(x) at every continuity point x of f.
Corollary: integration by parts.

From Lecture 17: Methods of integration. Integrability in elementary functions of rational functions (based on decomposition into simple fractions), and rational trigonometric functions (based on rational parameterization of the circle).
Here is the practice set on integration (and here are the answers )

From Lecture 18: Series
Definition of convergent series. The Cauchy criterion of convergence.
\lim a_n =0 as a ncessary condition for convergence for \sum a_n, but not sufficient
The integral criterion (of convergence of series of decreasing positive terms).
Examples: For which p, do the following series converge: \sum 1/n^p, \sum 1/n (\log n)^p, \sum 1/n \log n (\log \log n)^p, etc?
Definition of absolutely convergent series.
Thm: An absolutely convergent series converges, and its sum does not change under reordering.
Counter-examples (in the case of non-absolute conergence) based on alternating series: Convergence of an alternating series \sum (-1)^n a_n, assuming that decreasing positive a_n tend to 0, and dependence of the sum on the ordering of the terms, assuming that \sum a_n is infinite.

From Lecture 19: Comparison, Root, and Ratio Tests of (absolute) convergence. Example of the exponenntial series.

From Lecture 20: Power series: \sum_{n=0}^{\infty} a_n x^n. The formula for convergence radius R = 1/ \lim sup |a_n|^{1/n} .
Theorem about (sum and) product of absolutely convergent series, and its application to power series.
The multiplicative property of the exponential function.
Here is the practice set on series. ( Answers. 1: 1. 2: no. 3: yes. 4: show that 1/1+1/4+1/7+1/10+...=\infty. 5: compare to 1/2+1/3+1/4+... 6: (-4,4), diverges at both endpoints. 7: (-1,1); for \a > or = 0, converges absolutely at |x|=1; for -1< \a < 0, converges non-abslutely at x=1 and diverges as x=1; for \a < or = -1, diverges at |x|=1. 8: (-1,1), diverges at both endpoints. 9: converges only at x=0. 10: (n+1)(n+2)/2.

From Lecture 21: Sequences and series of funcions.
Counter-examples (from the opening section of Chapter 7 in Rudin's book) to independence of double limits on the order, to conservation of continuity, differentiability, integrability properties under passing to the limit, and to the possibility of termwise differentiation and integration of the sum of a series of functions.

From Lecture 22: Uniform convergence. Definition. Theorems about continuity, integration, and differentiation of uniformly convergent sequences and series of functions. Example: the derivative of \exp x.

From Lecture 23: Example: Existence of a continuous surjecive map [0,1] --> [0,1]^2 (Peano curve) or
Existence of continuous nowhere differeniable functions |R --> |R.

From Lecture 24: Equicontinuous families of functions. The Arzela-Ascoli Theorem (essentially pages 157-158 of the 3rd edition of Rudin's book).

From Lecture 25: The Weierstrass approximation theorem:
For every continuous function on [a,b], there is a sequence of polynomials which tends to this function uniformly on [a,b]. Proof: The first 2 pages from the section "The Stone-Weierstrass Theorem" in Rudin's text.
THIS MATERIAL CONCLUDES THE THEORETICAL COMPONENT OF OUR COURSE

Here is the set of problems for our semminar discussions on April 30, May 5, and May 7.
Make-up quiz on series is SCHEDULED for Thursday, April 30