Instructor: Matthew Cha
Lecture: MWF 11:30am-12:20p in A220 Wells Hall
Office Hours: MF 5p-6p; Tu 6p-7p in C217 Wells Hall
Email: chamatth "at" msu "dot" edu
Course grader: Leo Abbrescia
Email: abbresci "at" msu "dot" edu
This is a first course in a introductory two-semester sequence on real analysis. The topics we will cover include the real number system, sequences and series, limits of functions and continuity, differentiation, sequences and series of functions, and uniform convergence. The objectives of the course are to obtain a deep understanding of the theory underlying calculus and learn what constitutes a rigorous mathematical proofs and how to write one.
The official textbook is S. Abbott, Understanding Analysis, 2nd ed., (Springer, 2015). An online copy of the book is freely available to students via the MSU library. We will roughly cover chapters 1-6.
There will be 10 homework sets. Homeworks will typically be assigned a week before they are due. Late homework will not be accepted. Students are encouraged to discuss the homework assignments among themselves, but are expected to turn in their own work - copying someone else's is not acceptable. Homework scores will contribute 20% to the final grade. The lowest homework score will be dropped. It is recommended, although not required, that you type your homework solutions in LaTeX. Here is a template you can use: template.tex, template.pdf
There will be two in-class midterm exams. Exam 1 is set for Friday, February 15 and Exam 2 is set for Friday, April 5. The final is scheduled for Thursday, May 2 at 12:45p - 2:45p. There will be no makeup tests.
The course grade will be calculated based on the following percentages: Homework 20%, Exam 1 25%, Exam 2 25%, Final 30%. Grades will be posted on D2L.
| 7 January 19 |
§1.1. Rational numbers, real numbers and examples The irrationality of √2, law of cancellation, geometric series: S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2, divergent series: S = 1 + 2 + 4 + 8 + 16 + ... = -1 ? |
| 9 January 19 |
§1.2. Some preliminaries: sets and functions HW1 [solutions] (due Wed 16 Jan 19) |
| 11 January 19 |
§1.3. ℝ and the axiom of completeness least upper bound, greatest lower bound |
| 14 January 19 |
§1.4. Consequences of completeness nested interval property, Archimedean property |
| 16 January 19 |
§1.4 Consequences of completeness continued Density of rationals, existence of √2 §1.5 Cardinality HW2 [solutions] (due Wed 23 Jan 19) |
| 18 January 19 |
§1.5 Countable sets; ℚ |
| 23 January 19 |
§1.5 Uncountable sets; ℝ §1.6 Cantor's theorem for power sets and diagonal argument HW3 (due Fri 1 Feb 19) |
| 25 January 19 |
§2.2 Convergent sequences §2.3 Properties of convergent sequences Uniqueness of limits, boundedness |
| 28 January 19 |
§2.3 Properties of convergent sequences Limits preserve inequality, Squeeze Theorem, algebraic limit laws |
| 30 January 19 |
Classes cancelled due to weather, see MSU Today. HW4 (due Wed 6 Feb 19) |
| 1 February 19 | §2.4 Monotone Convergene Theorem |
| 4 February 19 | §2.5 Subsequences and Bolzano-Weierstrass |
| 6 February 19 |
§2.6 The Cauchy criterion HW5 (due Wed 13 Feb 19) |
| 8 February 19 |
§2.7 Infinite series and converge test Practice Exam 1 |
| 11 February 19 |
§2.7 Rearrangements Review of Practice Exam 1 |
| 13 February 19 | Review for Exam 1 |
| 15 February 19 | Exam 1 [solutions] |
| 18 February 19 | Review of Exam 1 |
| 20 February 19 |
§3.1 The Cantor Set §3.2 Open sets HW6 (due Wed 27 Feb 19) |
| 22 February 19 |
§3.2 Closed sets §3.3 Compact sets |
| 25 February 19 | §3.3 Compact sets |
| 27 February 19 | §4.1 Examples of Dirichlet and Thomae |
| 29 February 19 | §4.2 Functional limits |
| 11 March 19 | §4.2 Functional limits and their properties |
| 13 March 19 |
§4.2 Properties of functional limits §4.3 Continuous functions HW7 (due Wed 20 Mar 19) |
| 15 March 19 |
§4.3 Continuous functions and compositions |
| 18 March 19 |
§4.5 Continuous functions on intervals Extreme Value Theorem, Location of roots |
| 20 March 19 |
§4.5 Continuous functions on intervals Intermediate Value Theorem, Preservation theorems |
| 22 March 19 |
§4.5 Preservation theorems for continuous functions §4.4 Continuous functions on compact sets HW8 (due Mon 1 Apr 19) |
| 25 March 19 |
§4.4 Uniform continuity |
| 27 March 19 |
§ 4.4 Uniform continuity continued Lipschitz function, continuous extensions |
| 29 March 19 |
§ 4.6 Set of discontinuities one-sided limits, monotone functions Practice Exam 2 |
| 1 April 19 |
§ *Step functions and uniform approximation |
| 3 April 19 |
§ Exam 2 review |
| 5 April 19 |
§ Exam 2 [solutions] |
| 8 April 19 |
§ 5.2 Derivatives; examples and properties |
| 10 April 19 |
§ 5.2 Fermat's method and Darboux Theorem HW9 (due Wed 17 Apr 19) |
| 12 April 19 |
§ 5.3 The Mean Value Theorem and applications constant functions, l'Hopistal's Rule |
| 15 April 19 |
§ 6.2 Sequences of functions Pointwise and uniform convergence |
| 17 April 19 |
§ 6.2 Non-uniform convergence § 6.2 Continuous Limit Theorem § 6.3 Differentiable Limit Theorem HW10 (due Wed 26 Apr 19) |
| 19 April 19 |
§ 6.4 Series of functions, Weierstrass-M test § 6.5 Power series |
| 22 April 19 |
§ 6.5 *Radius of convergence, Abel's Theorem |
| 24 April 19 |
§ 6.6 *Taylor Series and remainder theorem |
| 26 April 19 |
Final exam review |
| 2 May 19 |
Final Exam [solutions] |
| * Material not covered on exams |