Instructor: Anton Izosimov
Textbook: Kenneth R. Davidson and Allan P. Donsig, Real Analysis and Applications: Theory in Practice
Tests and assignments:
- Quiz 1 -- the week of Jan 23, during your tutorial session; coverage
- Term Test 1 -- Feb 8, during class time; coverage; test; solutions
- Homework Assignment -- uploaded on Feb 15, due on Mar 1 at 2:10pm; solutions
- Term Test 2 -- Mar 8, during class time; coverage; test; solutions
- Quiz 2 -- the week of Mar 20, during your tutorial session; coverage
- Final Exam -- April 17; coverage; exam
Covered material and recommended exercises:
- Fri, Jan 6 -- introduction to real numbers, non-existence of a rational number whose square is 2 References: see, e.g., https://www.math.utah.edu/~pa/math/q1.html Recommended problems
- Wed, Jan 11 -- definition of real numbers, comparisson of real numbers, upper and lower bounds, the least upper bound principleTextbook references: Sections 2.2 and 2.3Recommended problems: Exercises H and I for Section 2.2, Exercise A for Section 2.3, and these problems
- Fri, Jan 13 -- proof of the least upper bound priniciple Textbook reference: Section 2.3 Recommended problems
- Wed, Jan 18 -- arithemtic operations with real numbers, examples of computations of sup and inf, construction of the square root of 2, introduction to limits Recommended problems: Exercises D(a) and D(b) for Section 2.3, and these problems Solutions to Problems 1 and 2 from the PDF file
- Fri, Jan 20 -- definition of the limit of a sequence Textbook reference: Section 2.4 Recommended problems: Exercises A(a), A(b), B, E, G for Section 2.4
- Wed, Jan 25 -- examples of limits, squeeze theorem, basic properties of limits, monotone convergence theorem Textbook reference: Sections 2.4, 2.5, 2.6 Recommended problems: prove the squeeze theorem, Exercises A(c-d-e), C, D, F, H, J, K for Section 2.4, Exercises A, B, C, E, G, I, J for Section 2.5, Exercises A, B for Section 2.6 Solutions to Exercise J for Section 2.4 and Exercises C and J for Section 2.5
- Fri, Jan 27 -- nested intervals lemma, subsequences, Bolzano-Weierstrass theorem Textbook reference: Sections 2.6, 2.7 Recommended problems: Exercise D for Section 2.7 and these problems Solution of Exercise D for Section 2.7 and Question 2 from the PDF file
- Wed, Feb 1 -- Cauchy sequences, equivalence of being Cauchy and being convergent Textbook reference: Section 2.8 Recommended problems: Exercises A, B, E, F for Section 2.8, and these problems
- Fri, Feb 3 -- limits of functions and continuity (for functions of one variable) Textbook reference: Section 5.1 Recommended problems: Exercise C for Section 5.1 and these problems Solution of Problems 1b and 1c from the PDF file
- Fri, Feb 10 -- basic properties of continuous functions, including continuity criterion in terms of sequences Textbook reference: Section 5.3 Recommended problems: Exercise H for Section 5.3 and these problems Solution to Exercise H
- Wed, Feb 15 -- intermediate value theorem, monotone functions Textbook reference: Sections 5.6, 5.7 Recommended problems (not to be confused with the problem set which you will hand in): Exercises A, B, C, D, G for Section 5.6 (here you can use, without proof, that triginometric functions are continous on their domains; you can also use differentiation), Exercises A (here decreasing and monotone means strictly decreasing and stricly monotone), B (here you need to prove that f is strictly monotone), C for Section 5.7 and these problems Solution to Exercise C for Section 5.6 and Problem 1 from the PDF file
- Fri, Feb 17 -- a continous function on a closed interval is bounded and attains its supremum and infinum Textbook reference: Section 5.4 Recommended problems: Exercises B, C, H, K for Section 5.4
- Wed, Mar 1 -- differentiation, proof of the chain rule, Fermat's theorem, Rolle's theorem, mean value theorem Textbook reference: Sections 6.1, 6.2 Recommended problems: Exercises A, D, E, F, H for Section 6.1, Exercises A, B, C for Section 6.2, and these problems Solutions to problems from the PDF file
- Fri, Mar 3 -- definition of the Riemann integral Reference: Section 1 of these lecture notes (the textbook discusses Riemann integration in Section 6.3, but their definitions are slighly different, although equivalent to ours) Recommended problems: This problem, and Exercises D, H, I for Section 6.3
- Fri, Mar 10 -- uniform continuity, a function continuous on [a,b] is uniformly continous on [a,b] Reference: Section 5.5 Recommended problems: Exercises A, F, H for Section 5.5, and these problems
- Wed, Mar 15 -- integrability of continuous functions Reference: Section 2 of these lecture notes (the textbook discusses this material in Section 6.3, but their definitions are slighly different, although equivalent to ours) Recommended problems: These problems, and Exercises B, K, L for Section 6.3 Solution to Problem 6 from the PDF file
- Fri, Mar 17 -- fundamental theorem of calculus Reference: Section 6.4 Recommended problems: Exercises C, D for Section 6.4 and these problems Solution to Problem 2 from the PDF file
- Wed, Mar 22 -- metric spaces, examples, convergence in metric spaces, uniform versus pointwise convergence for sequences of functions, Cauchy sequences in metric spaces, complete metric spaces Reference: Sections 9.1 and 8.1 Recommended problems: Exercise J for Section 9.1 and these problems Solution to Problem 7 from the PDF file
- Fri, Mar 24 -- completeness of the metric space of continuous functions with the uniform metric Reference: Section 8.2 Recommended problems: Exercises Aa, C, D, I for Section 8.2 and these problems Solution to Problem 1 from the PDF file
- Wed, Mar 29 -- continuous mappings between metric spaces, contraction mapping principle Reference: Sections 9.1 and 11.1 Recommended problems: these problems
- Fri, Mar 31 -- introduction to first order differential equations Reference: see, for instance, Section 9.1 of Stewart's textbook on single variable calculus Recommended problems: these problems
- Wed, Apr 5 -- proof of the existence and uniqueness theorem for first order differential equations