MATHS 471: Real Analysis 1 (3)

MATHS 571: Real Analysis 1 (3)
Syllabus
1. Prerequisite: MATHS 215 and 267, or permission of department chairperson.
Note: Not open to students who have credit in MATHS 471.
2. Course Description: Properties of the real numbers. Cardinality. Topological properties of metric spaces:
compactness, completeness, connectedness. Sequences and series. Continuous functions. Differential
calculus of real- and vector-valued functions of one real variable.
3. Course Objectives: The main objective of the course is a careful and rigorous treatment of the main ideas of
differential calculus, as first encountered by the student in a freshman-level calculus course. The concepts of
sequences in metric spaces and numerical series will also be studied. MATHS 571 includes a basic
treatment of metric spaces, providing the necessary technical language and framework for the discussion of
modern analysis topics. Students completing the course will gain a deeper understanding of differential
calculus and they will gain experience in writing proofs and solving problems related to the content of the
course.
4. Course Rationale: This course primarily serves Mathematics majors and graduate students (since the course
is taught with MATHS 471). Students who take this course will have a better understanding of the
theoretical underpinnings of differential calculus. Sequences and infinite series will be treated in depth.
Since the course provides essential knowledge for further study in analysis, topology, geometry, and applied
mathematics, it is an essential course for students contemplating earning a doctorate in the mathematical
sciences.
5. Course Content: Properties of the real numbers, including completeness and least upper bound property;
cardinality of sets; theory of metric spaces, with emphasis on Euclidean space, including open and closed
sets, Cauchy and convergent sequences, compactness, completeness, and connectedness; numerical
sequences and infinite series, including convergence tests, rearrangement, and multiplication of series;
continuous functions between metric spaces, including the Extreme Value Theorem, Intermediate Value
Theorem, and uniform continuity; differentiability of functions of one variable, including the product rule,
quotient rule, chain rule, Mean Value Theorem, continuity of derivatives, l’Hospital’s Rule, and Taylor’s
theorem.
6. Course Format: Lecture/discussion. The amount of material to be covered does not allow for complete
proofs to be given in class of all the results which are important. The instructor is responsible for choosing a
representative sample of proofs that illustrate the most common techniques. Students are expected to
understand other proofs through careful reading of the text.
7. Methods of Evaluating Student Performance: Typical components are homework assignments, projects,
take-home exams (possibly involving group work), in-class exams, student presentations, and oral exams.
The evaluation and weight of these components are at the discretion of the instructor. Students may be
evaluated on their understanding of the content material of the course, their ability to solve problems related
to the course material, and on their ability to communicate mathematically to others orally and/or in
writing.
Although this course may be taught with an undergraduate course, students earning graduate credit for this
course will complete work and be evaluated on that work appropriate to graduate study. Specifically, these
students will be expected to (i) demonstrate comprehension and proper use of terminology; (ii) attain
proficiency in formal logic, foundational theory, and proof writing; (iii) master core graduate level

computation skills.
8. Evaluation of the Course: The instruction of the course is evaluated by departmental student evaluations and
peer evaluation. The course is reviewed and revised periodically by the Departmental Graduate Programs
Committee.
9. Addendum: Suitable textbooks for this course include the following.
Rudin, W., Principles of Mathematical Analysis (3 rd Edition)
R. Stankewitz, 10/10/09 (Adapted from Maths 471 syllabus of Ahmed Mohammed, Michael Karls, Spring
2007); R. Stankewitz, 2/15/12;