Fall 2025
- MATH 409-503: Analysis on the Real Line
Time and venue: MWF 11:30 a.m.–12:20 p.m., BLOC 160
In-person office hours (BLOC 301b):
- MWF 2:00–3:00 p.m.
- or by appointment
Online office hours (for a link, go to Canvas):
- Thursday 5:00–6:00 p.m.
- or by appointment
Online office hours during the finals (for a link, go to Canvas):
- Wednesday, December 10, 2:00–3:00 p.m.
- Thursday, December 11, 5:00–6:00 p.m.
- or by appointment
In-person office hours during the finals (BLOC 301b):
- Friday, December 12, 11:00 a.m.–2:00 p.m.
- Monday, December 15, 11:00 a.m.–2:00 p.m.
Test 1: Wednesday, October 8 (Sample problems)
Final exam: Tuesday, December 16, 10:30 a.m.–12:30 p.m.
Course outline:
Part I: Axiomatic model of the real numbers
- Axioms of an ordered field
- Completeness axiom
- Principle of mathematical induction
- Countable and uncountable sets
Thomson/Bruckner/Bruckner: Chapter 1, Appendix A, Section 2.3
Lecture 1: Axioms of a field.
Lecture 2: Properties of ordered fields.
Lecture 3: Supremum and infimum. Completeness axiom.
Lecture 4: Archimedean property (continued). Density of rational numbers. Existence of square roots.
Lecture 5: Existence of square roots (continued). Mathematical induction. Binomial formula.
Lecture 6: Functions. Countable and uncountable sets.
Lecture 7a: Absolute value. Metric spaces.
Part II: Sequences and infinite sums
- Limits of sequences
- Bolzano-Weierstrass theorem
- Cauchy sequences
- Convergence of series
- Tests for convergence
- Absolute convergence
Thomson/Bruckner/Bruckner: Chapters 2-3
Lecture 7b: Limit of a sequence.
Lecture 8: Properties of limits. Divergent sequences.
Lecture 9: Algebra of limits.
Lecture 10: Monotonic sequences.
Lecture 11: More examples of limits.
Lecture 12: Bolzano-Weierstrass theorem. Cauchy sequences.
Lecture 13: Limit points. Upper and lower limits.
Lecture 14: Convergence of infinite series.
Lecture 15: Ratio and root tests for convergence.
Lecture 16: More tests for convergence.
Lecture 17: Absolute convergence.
Lecture 18: Review for Test 1.
- T/B/B 1.1-1.10, 2.1-2.13, 3.1-3.2, 3.4-3.7
Part III: Continuity
- Topology of the real line
- Limits of functions
- Continuouos functions
- Uniform continuity
Thomson/Bruckner/Bruckner: Chapters 4-5
Lecture 19: Topology of the real line: classification of points.
Lecture 20: Topology of the real line: open and closed sets.
Lecture 21: Compact sets. Limit of a function.
Lecture 22: Limit of a function (continued).
Lecture 23: Limits of trigonometric functions. Continuity.
Lecture 24: More on continuous functions.
Lecture 25: Points of discontinuity. Monotonic functions.
Lecture 26: Exponential function. Uniform continuity.
Part IV: Differential and integral calculus
- Differentiability, properties of the derivative
- The mean value theorem
- Taylor's theorem
- Riemann sums, the Riemann integral
- The fundamental theorem of calculus
Thomson/Bruckner/Bruckner: Chapters 7-8
Lecture 27: The derivative. Differentiability theorems.
Lecture 28: Differentiability theorems (continued). Derivatives of elementary functions.
Lecture 29: An important limit. Mean value theorem.
Lecture 30: Mean value theorem (continued). L'Hopital's rule.
- T/B/B 7.4, 7.6, 7.9, 7.11
Lecture 31: L'Hopital's rule (continued). Taylor's formula.
Lecture 32: Review for Test 2.
- T/B/B 4.1-4.6, 5.1-5.2, 5.4-5.9, 7.1-7.7, 7.9, 7.11-7.12
Lecture 33: Convex functions. Riemann sums.
Lecture 34: Darboux sums. Integrable functions.
Lecture 35: Properties of the integral.
Lecture 36: Fundamental theorem of calculus. Indefinite integral.
Lecture 37: Integration by parts. Integration by substitution.
Lecture 38: Improper Riemann integrals.
Lecture 39: More on the integral. Jensen's inequality.
Lecture 40: Review for the final exam.
- T/B/B 1.1-1.10, 2.1-2.13, 3.1-3.2, 3.4-3.7, 4.1-4.6, 5.1-5.2, 5.4-5.9, 7.1-7.7, 7.9-7.12, 8.1-8.8