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 | 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

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)