MAT2125 – Elementary Real Analysis

Course Outline

Instructor: Patrick Boily, uOttawa

Questions related to the course content should be asked on the course’s Piazza forum (that space is shared with the other section). Section-specific communications with the instructor should take place on the course’s Slack workspace (the registration link is available in the course kick-off email sent on January 9 and in the “Announcements” section on Brightspace).

Exercise Sessions Schedule:
   Wednesdays, 13:00-14:20, Jan 13 – Apr 17
   Wednesdays, 14:30-15:50, Jan 13 – Apr 07
   Fridays, 11:30-12:50, Jan 15 – Apr 13
   No session on Feb 17, Feb 19, and Apr 02

Exercise Sessions Zoom link

Important Dates:
   Jan 29: assignment 1 due at midnight
   Feb 12: assignment 2 due at midnight
   Mar 01: assignment 3 due at midnight
   Mar 03: midterm examination at 14:30
   Mar 26: assignment 4 due at midnight
   Apr 14: assignment 5 due at midnight

Note: the assignments must be typeset using LaTeX and uploaded as a single PDF document to Brightspace.

A. Savage’s Course Notes

Textbooks:
   J. Lebl’s Basic Analysis I and Basic Analysis II
   W.F. Trench’s Introduction to Real Analysis
   E. Borman & R. Rogers’s Real Analysis
   P. Keefe & D. Guichard’s Introduction to Higher Mathematics

Course Documents:
   Problem Set (in-class exercises, the complete list in now available)
   Some numbered results (to accompany A. Smith’s video lectures)
   Their proofs (updated)
   A. Smith’s Introduction to LaTeX
   A. Smith’s Field and Order Axioms — Notes
   Midterm Exam Solutions
   Final Exam Solutions

Assignments:
   Assignment 1 (solutions)
   Assignment 2 (solutions)
   Assignment 3 (solutions)
   Assignment 4 (solutions)
   Assignment 5 (solutions)

Some of the video lectures are available without a soundtrack here. In case of divergence, the videos on the website (this page) take precedence.

State of the course address, after the first two months (30:00)

Topics:
Classical problems; calculus as an informal collection of solutions; the need for formalism. How to set up LaTeX for assignments.

Video Lectures (0:37:33)
   Jan 11-Jan 17: 1.1 Historical Perspective (22:11)
   Jan 11-Jan 17: 1.2 Introduction to LaTeX (15:22)

Topics:
Field and order axioms; mathematical induction; Archimedean property; lower/upper bound; infimum/supremum; completeness axiom; constructing and finding reals. Useful inequalities: triangle; Bernoulli; Cauchy-Schwartz. Finite sets; infinite sets; countable sets; uncountable sets; infinite subsets; countability of Q; uncountability of [0,1]; uncountability of R. Nested intervals theorem.

Video Lectures (2:41:25)
   Jan 11-Jan 17: 2.1 Number Systems and Properties of R | Part 1 (28:39)
   Jan 11-Jan 17: 2.1 Number Systems and Properties of R | Part 2 (09:03)
   Jan 11-Jan 17: 2.1 Number Systems and Properties of R | Part 3 (13:00)
   Jan 18-Jan 24: 2.1 Number Systems and Properties of R | Part 4 (15:26)
   Jan 18-Jan 24: 2.1 Number Systems and Properties of R | Part 5 (06:56)
   Jan 18-Jan 24: 2.1 Number Systems and Properties of R | Part 6 (06:11)
   Jan 18-Jan 24: 2.2 Cardinality | Part 1 (23:28)
   Jan 18-Jan 24: 2.2 Cardinality | Part 2 (19:51)
   Jan 18-Jan 24: 2.2 Cardinality | Part 3 (12:55)
   Jan 25-Jan 31: 2.3 Nested Interval Theorem | Part 1 (08:58)
   Jan 25-Jan 31: 2.3 Nested Interval Theorem | Part 2 (16:58)

Problem Set Exercises: Q1-Q18

Study Aid – Chapter 2

Topics:
Sequences; limit; boundedness. Arithmetic of limits; squeeze theorem for limits. Monotone sequences; bounded monotone convergence theorem. Subsequences; subsequences of convergent sequences; Bolzano-Weierstrass theorem; boundedness and convergence of subsequences. Cauchy sequences; boundedness of Cauchy sequences; equivalence of convergence and Cauchy sequences in R. Euclidean spaces; norms; sequences and limits in R^d. Topology of R^d; open balls; complements; open sets; closed sets; interior points; isolated points; accumulation points; boundary points. Compactness; open cover; Heine-Borel theorem; sequential compactness.

Video Lectures (3:39:54)
   Jan 25-Jan 31: 3.1 Infinity vs Intuition (07:06)
   Jan 25-Jan 31: 3.2 Introduction to Sequences | Part 1 (17:57)
   Jan 25-Jan 31: 3.2 Introduction to Sequences | Part 2 (07:05)
   Jan 25-Jan 31: 3.3 Calculating Limits (15:26)
   Jan 25-Jan 31: 3.4 Monotone Sequences | Part 1 (07:23)
   Feb 01-Feb 07: 3.4 Monotone Sequences | Part 2 (13:35)
   Feb 01-Feb 07: 3.5 Bolzano-Weierstrass Theorem | Part 1 (09:41)
   Feb 01-Feb 07: 3.5 Bolzano-Weierstrass Theorem | Part 2 (24:06)
   Feb 01-Feb 07: 3.5 Bolzano-Weierstrass Theorem | Part 3 (02:58)
   Feb 01-Feb 07: 3.6 Cauchy Sequences | Part 1 (13:05)
   Feb 01-Feb 07: 3.6 Cauchy Sequences | Part 2 (26:58)
   Feb 08-Feb 14: 3.7 Sequences and Topology of R^d | Part 1 (13:39)
   Feb 08-Feb 14: 3.7 Sequences and Topology of R^d | Part 2 (07:36)
   Feb 08-Feb 14: 3.7 Sequences and Topology of R^d | Part 3 (26:04)
   Feb 08-Feb 14: 3.7 Sequences and Topology of R^d | Part 4 (27:15)

Problem Set Exercises: Q19-Q44

Study Aid – Chapter 3

Topics:
Limit points in R (reprise); link between limit points and convergent sequences; limit of a function; uniqueness of limits; sequential definition of limits. Properties of limits; squeeze theorem for functions. Continuous functions; continuity and elementary operations; composition of continuous functions; continuous image of a compact set, uniform continuity. Intermediate value theorem; fixed point theorem; Max/Min theorem; continuous image of a closed and bounded interval.

Video Lectures (2:15:43)
   Feb 22-Feb 28: 4.1 Limit of a Function | Part 1 (12:49)
   Feb 22-Feb 28: 4.1 Limit of a Function | Part 2 (16:49)
   Feb 22-Feb 28: 4.1 Limit of a Function | Part 3 (18:34)
   Feb 22-Feb 28: 4.1 Limit of a Function | Part 4 (14:38)
   Feb 22-Feb 28: 4.2 Properties of Limits | Part 1 (07:32)
   Feb 22-Feb 28: 4.2 Properties of Limits | Part 2 (11:08)
   Mar 01-Mar 07: 4.3 Continuous Functions | Part 1 (14:09)
   Mar 01-Mar 07: 4.3 Continuous Functions | Part 2 (10:35)
   Mar 01-Mar 07: 4.4 Images and Continuity (08:26)
   Mar 01-Mar 07: 4.5 Intermediate Value Theorem (11:41)
   Mar 01-Mar 07: 4.6 Maximum/Minimum Theorem (06:51)
   Mar 01-Mar 07: 4.7 Uniform Continuity (10:10)

Problem Set Exercises: Q45-Q65

Study Aid – Chapter 4

Topics:
Definition of the derivative; differentiability and continuity; differentiation rules; Carathéodory’s theorem; chain rule. Relative extrema; Rolle’s theorem; mean value theorem; functions with constant derivatives. Higher-order derivatives; Taylor’s theorem. Monotone functions; 1st derivative test; change of sign of the derivative; Darboux’ theorem. Integrals as areas; integral of a continuous function over a closed and bounded interval; partitions and refinements; lower/upper sums; Riemann-integrable functions; conventions; Riemann’s criterion; familiar formulas. Riemann-integrability of monotone functions.
Properties of Riemann integrals: additivity; linearity; triangle inequality; comparisons; mean value theorem for integrals; Riemann-integrable fonction with uncountable discontinuities. Composition theorem. 1st fundamental theorem of calculus; 2nd fundamental theorem of calculus. Integration by parts; 1st substitution theorem; 2nd substition theorem; squeeze theorem for integrals.

Video Lectures (4:18:56)
   Mar 08-Mar 14: 5.1 Differentiation | Part 1 (09:14)
   Mar 08-Mar 14: 5.1 Differentiation | Part 2 (16:22)
   Mar 08-Mar 14: 5.1 Differentiation | Part 3 (12:42)
   Mar 08-Mar 14: 5.2 Mean Value Theorem | Part 1 (15:45)
   Mar 08-Mar 14: 5.2 Mean Value Theorem | Part 2 (10:02)
   Mar 08-Mar 14: 5.3 Taylor’s Theorem | Part 1 (20:41)
   Mar 08-Mar 14: 5.3 Taylor’s Theorem | Part 2 (07:55)
   Mar 15-Mar 21: 5.4 Relative Extrema and Derivatives | Part 1 (07:48)
   Mar 15-Mar 21: 5.4 Relative Extrema and Derivatives | Part 2 (13:20)
   Mar 15-Mar 21: 5.5 Riemann Integration | Part 1 (14:41)
   Mar 15-Mar 21: 5.5 Riemann Integration | Part 2 (16:57)
   Mar 15-Mar 21: 5.5 Riemann Integration | Part 3 (04:52)
   Mar 15-Mar 21: 5.6 General Integration Results | Part 1 (08:26)
   Mar 15-Mar 21: 5.6 General Integration Results | Part 2 (17:48)
   Mar 22-Mar 28: 5.6 General Integration Results | Part 3 (04:45)
   Mar 22-Mar 28: 5.6 General Integration Results | Part 4 (03:16)
   Mar 22-Mar 28: 5.6 General Integration Results | Part 5 (11:46)
   Mar 22-Mar 28: 5.7 Composition Theorem | Part 1 (15:43)
   Mar 22-Mar 28: 5.7 Composition Theorem | Part 2 (01:13)
   Mar 22-Mar 28: 5.8 Fundamental Theorem of Calculus | Part 1 (10:42)
   Mar 22-Mar 28: 5.8 Fundamental Theorem of Calculus | Part 2 (14:07)
   Mar 22-Mar 28: 5.8 Fundamental Theorem of Calculus | Part 3 (08:14)
   Mar 22-Mar 28: 5.9 Evaluation of Integrals (12:37)

Problem Set Exercises: Q66-Q87

Study Aid – Chapter 5

Topics:
Sequences of functions; pointwise convergence; uniform convergence; Cauchy property for sequences of functions. Limit interchange theorems; sequences of uniformly continuous functions; sequences of differentiable functions; sequences of Riemann-integrable functions.

Video Lectures (1:10:24)
   Mar 29-Apr 04: 6.1 Pointwise and Uniform Convergence | Part 1 (20:50)
   Mar 29-Apr 04: 6.1 Pointwise and Uniform Convergence | Part 2 (14:18)
   Mar 29-Apr 04: 6.2 Limit Interchange Theorems | Part 1 (07:15)
   Mar 29-Apr 04: 6.2 Limit Interchange Theorems | Part 2 (14:48)
   Mar 29-Apr 04: 6.2 Limit Interchange Theorems | Part 3 (13:13)

Problem Set Exercises: Q88-Q92

Study Aid – Chapter 6

Topics:
Series of real numbers; Cauchy criterion for series; comparison test; alternating series test; ratio test; root test; absolute convergence; rearrangement. Series of functions; convergence; Cauchy criterion; Weierstrass M-test; Abel’s test. Power series; radius of convergence; Fourier series.

Video Lectures (1:16:41)
   Apr 05-Apr 11: 7.1 Series of Real Numbers | Part 1 (09:05)
   Apr 05-Apr 11: 7.1 Series of Real Numbers | Part 2 (12:12)
   Apr 05-Apr 11: 7.1 Series of Real Numbers | Part 3 (12:11)
   Apr 05-Apr 11: 7.2 Series of Functions (11:29)
   Apr 05-Apr 11: 7.3 Power Series | Part 1 (13:24)
   Apr 05-Apr 11: 7.3 Power Series | Part 2 (11:38)
   Apr 05-Apr 11: 7.3 Power Series | Part 3 (06:49)

Problem Set Exercises: Q93-Q100

Additional Videos (some overlap with chapter 6):
   pointwise convergence (11:01)
   uniform convergence (15:26)

Study Aid – Chapter 7