MAT2125 – Elementary Real Analysis

Course Details	Important Documents
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 P. Keefe & D. Guichard’s Introduction to Higher Mathematics Course Documents: Problem Set (in-class exercises, the complete list in now available) A few analysis proofs (updated) Some numbered results (to accompany A. Smith’s video lectures) A. Smith’s Introduction to LaTeX A. Smith’s Field and Order Axioms — Notes Midterm Exam Solutions Assignments: Assignment 1 (solutions) Assignment 2 (solutions) Assignment 3 (solutions) Assignment 4 Assignment 5 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)
1. Preamble	2. The Real Numbers
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
3. Sequences	4. Limits and Continuity
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	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
5. Calculus	6. Sequences of Functions
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 Mar 08-Mar 14: Mar 08-Mar 14: Mar 08-Mar 14: Mar 08-Mar 14: Mar 08-Mar 14: Mar 08-Mar 14: Mar 15-Mar 21: 5.4 Relative Extrema Mar 15-Mar 21: Mar 15-Mar 21: Mar 15-Mar 21: Mar 15-Mar 21: Mar 15-Mar 21: Mar 15-Mar 21: Mar 22-Mar 28: 5.6 General Integration Mar 22-Mar 28: Mar 22-Mar 28: Mar 22-Mar 28: Mar 22-Mar 28: Mar 22-Mar 28: Mar 22-Mar 28: Mar 22-Mar 28: Mar 22-Mar 28: \| Part 1 (09:14) 5.1 Differentiation \| Part 2 (16:22) 5.1 Differentiation \| Part 3 (12:42) 5.2 Mean Value Theorem \| Part 1 (15:45) 5.2 Mean Value Theorem \| Part 2 (10:02) 5.3 Taylor’s Theorem \| Part 1 (20:41) 5.3 Taylor’s Theorem \| Part 2 (07:55) and Derivatives \| Part 1 (07:48) 5.4 Relative Extrema and Derivatives \| Part 2 (13:20) 5.5 Riemann Integration \| Part 1 (14:41) 5.5 Riemann Integration \| Part 2 (16:57) 5.5 Riemann Integration \| Part 3 (04:52) 5.6 General Integration Results \| Part 1 (08:26) 5.6 General Integration Results \| Part 2 (17:48) Results \| Part 3 (04:45) 5.6 General Integration Results \| Part 4 (03:16) 5.6 General Integration Results \| Part 5 (11:46) 5.7 Composition Theorem \| Part 1 (15:43) 5.7 Composition Theorem \| Part 2 (01:13) 5.8 Fundamental Theorem of Calculus \| Part 1 (10:42) 5.8 Fundamental Theorem of Calculus \| Part 2 (14:07) 5.8 Fundamental Theorem of Calculus \| Part 3 (08:14) 5.9 Evaluation of Integrals (12:37) Problem Set Exercises: Q66-Q87	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
7. Series	Exercise Sessions
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 (TBD) 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 \| Part 1 (coming soon) Apr 05-Apr 11: 7.2 Series of Functions \| Part 2 (coming soon) Apr 05-Apr 11: 7.3 Power Series \| Part 1 (coming soon) Apr 05-Apr 11: 7.3 Power Series \| Part 2 (coming soon) Apr 05-Apr 11: 7.3 Power Series \| Part 3 (coming soon) Problem Set Exercises: Q93-Q100	Jan 15: Q2-Q4 + Archimedean Property Variants (1:02:39), [PDF] Jan 20: Q5-Q7 (0:45:58), [PDF (including Q8)] Jan 22: Q10-Q12 (0:51:31), [PDF] Jan 27: Q13-Q15 (0:45:14) [PDF] Jan 29: Q16-Q18, Q20 (0:43:39) [PDF (including Q19)] Feb 03: Q21-Q23, Q26 (0:57:59) [PDF (including Q24, Q25)] Feb 05: Q27-Q30 (0:53:11) [PDF] Feb 10: Q31-Q34 (1:02:31) [PDF] Feb 12: Q35-Q37 (42:39) [PDF] Feb 24: Q38-Q43 (1:11:43) [PDF] Feb 26: Q44-Q47 (1:06:50) [PDF (including Q48)] Mar 03: Q49-Q53 (no video today) [PDF] Mar 05: Q55-Q59 (1:02:09) [PDF (including Q54, Q60)] Mar 10: Q61-Q65 (00:00) Mar 12-Mar 18: Q66-Q72 (00:00) Mar 19-Mar 25: Q73-Q79 (00:00) Mar 26-Apr 01: Q80-Q87 (00:00) Apr 02-Apr 08: Q88-Q92 (00:00) Apr 09-Apr 15: Q93-Q100 (00:00)