MA 425: Real Analysis syllabus

Scope and Objectives

In this course, we will explore the theoretical underpinnings of the classical single-variable calculus. We will prove the familiar theorems upon which calculus runs. We will encounter a number of new objects, and we will also catch some old objects behaving in more subtle ways than one might expect from calculus. Welcome to the edge of the jungle.

The official course description is:

Real number system, functions and limits, topology on the real line, continuity, differential and integral calculus for functions of one variable. Infinite series, uniform convergence.

A student who successfully completes this course will (among other things):

  • Be able to state the precise definitions and compute/verify examples and nonexamples of: infimum and supremum, limits inferior and superior, limits of sequences, open and closed sets, limits of functions, limits of sequences of functions, continuous function, derivative, Riemann integrals,  radius of convergence.
  • Recall the precise statements of and apply the following theorems of single-variable calculus:  Weierstraß' Extreme Value Theorem, Bolzano's Intermediate Value Theorem, the Mean Value Theorems, L'Hôpital's Rules, Taylor's Theorem, the Fundamental Theorems of Calculus.
  • Recall the precise statements and apply the following theorems about the topology of the real line: Monotone Convergence Theorem, Bolzano-Weierstraß, Cauchy's Criterion, Fixpunktsäze, Sequential Characterizations.
  • Understand, analyze, and explain the proofs of all of the above theorems.
  • Synthesize the above theorems into proofs of new theorems.
  • Understand and explain the logical relationships between facts about the topology of the real line and the theorems of single-variable calculus.
  • Encounter and develop examples of sets and functions behaving counterintuitively.

The prerequisite is MA 225. From that course we will need the following concepts: mathematical induction, sets, functions. We will also need the following competencies: writing coherent proofs and carefully reading proofs and novel definitions. You will also need to be familiar with all the theorems of single-variable calculus.

Classroom Meetings

There are two kinds of classroom meetings: lectures and weekly problem sessions. All class meetings --- including problem sessions --- are essential and mandatory. If you do not attend it is likely that you will not pass the course.

Assignments and Grades

Your grade will be determined solely on the basis of your work on the assignments described here.

In this course we will use a variant of what is called specifications grading. Rather than achieving a certain percentage of available points, you will need to meet various criteria as described below. If you meet all the criteria for a certain letter grade, you will receive that letter grade. If you fail to complete the `C' specifications you will receive a grade of `D' or `F', depending on how many of the C specifications you have completed.

Missed Assignments If you will miss an exam, please let me know in writing and as soon as possible. I will be the sole arbiter of what constitutes a valid reason for missing an assignment, and of determining how missed assignments are to be made up, consistent with the University's policy on attendance, Regulation 02.20.03.

Preticipation I will occasionally assign a form of homework called preticipation. Each preticipation assignment consists reading/viewing and some exercises, to be completed prior to class. The reading/viewing is an essential part of the assignment.
Usually I will collect preticipation via Google form; sometimes I will ask you to bring something on paper. Preticipation will be graded according to whether you have made a good-faith effort to complete the exercise, rather than on correctness.

Reading and Reflection In addition to traditional homework, you will be asked to complete occasional reading and reflection assignments. These are short reflections on the course content that you will maintain in a Google Document. As with preticipation, reading and reflection assignments are graded on a good-faith effort to complete the exercise.

Quizzes Every week we will have a definition quiz (not a vocabulary quiz!). You will have a few (less than 3) minutes to recall a definition or the statement of a named theorem.

Homework Homework is the component of the course where you will learn the most --- by far. Homework problems will be assigned regularly, approximately 10 proofs per week. See below for administrative details.

Exams The two midterm exams will be closed-book and closed-notes. Most of the exams will consist of short-answer, matching, multiple choice, fill-in-the-blank, and reading comprehension type problems. You will be required to write some proofs on exams.

Your Rights as a Learner

In this course, you have the following rights:

  1. to be confused
  2. to make a mistake and to revise your thinking
  3. to ask questions
  4. to point out my mistake

In addition to the above rights, The course will be administered in accordance with all University Policies, Rules, and Regulations (some of which were already referenced above). Students are responsible for reviewing the NC State University PRR which pertain to their course rights and responsibilities.

Academic Integrity The definition of academic integrity is simple and broad:  do not take credit for others' work. This applies to all assignments. All assignments, absent an explicit statement to the contrary, should be completed individually. You may not collaborate on exams in any form. You may not use any aids except those approved through the Disability Services Office and arranged with me in advance. This includes, but is not limited to: textbooks, crib sheets, electronic calculators, electronic communications devices, tattoos of formulæ, and séances with the ghost of A-L Cauchy. Infractions against academic integrity will be addressed through the University's Office of Student Conduct pursuant to University Policy 11.35.01 and Regulation 11.35.02.

Outside of Class

Office Hours There is no need to make an appointment during my scheduled office hours. In addition, I am also available by appointment (email is best). Please come with a specific question or questions so that everyone's time is used efficiently. You may also consider using the office hours of other sections' instructors --- with their permission, of course.

Email All email announcements and correspondence will be sent to your official email address. I try to respond to all student emails regarding administrative matters within 24 hours --- if I have not responded within that time, please ask again, as your email may have been buried in my inbox.

Other Help I encourage you to make use of resources from other sources to get a different perspective. I will post some links to the course website.

Homework Policy

Marks for proofs. Each proof will receive one of the following marks:

S ("satisfactory"): this denotes a relatively high level of performance (maybe A-). These proofs meet all of the formatting requirements for a proof, are mathematically correct, are detailed, and prove what they claim to prove.
P ("progressing"): these proofs lack some of the characteristics for "satisfactory" but are on the right track and are amenable to revision to make them "satisfactory".
U ("unsatisfactory"): these proofs have significant, uncorrectable flaws. It is not worth your time to attempt a revision.

Only proofs marked S count for anything.

Collaboration. I encourage you to work together with your colleagues in the course, but to make this an effective learning tool, we will observe the following rules:

  • You may collaborate with at most two other students on each homework assignment.
  • If you choose to collaborate with other students on homework, each of you must submit a separate
    writeup. You must list the names of all collaborators in your submission.
  • If you consult a source other than the course notes, course textbook, or Dr. Cooper, you must list it
    in your submission.

    • Don't use sources you'd be embarrassed to cite.
  • Searching for solutions to the homework problems via the internet, other textbooks, etc., is expressly prohibited.
    • This does not mean you cannot seek clarification from an outside source on a point of confusion.
    • It is usually quite easy for me to tell when you've copied from somewhere.
    • Failure to adhere to these rules will result, at minimum, in not receiving a grade for the assignment in question. More serious infractions will incur more serious penalties.
  • There is no collaboration allowed on resubmission.

Resubmission. Proofs marked P may be revised and resubmitted:

  • You must resubmit a P proof within one week of receiving the original graded proof.
  • Revised proofs must be rewritten in their entirety.
  • Accompanying the revised proof, you must include the original submission, the revised version, and a reflective paragraph which addresses:
    • What did you change? Why did it need to be changed?
    • Why did you think your original attempt would work?
    • What lesson(s) can you apply to future proofs?
  • Only proofs may be revised and resubmitted.
  • There is no collaboration allowed on resubmission.
  • Revised proofs, once graded, can be revised again.

Formatting and style. If your homework does not meet the following formatting criteria, I may return it to you ungraded and require you to revise it before I grade it.

  • Handwriting should be legible (or submit typed homework via \LaTeX).
  • There should be sufficient whitespace, both at the margins and between proofs, for me to write comments.
  • Use only one side of the paper.
  • No strike-outs, caret insertions, or visible erasures are allowed. Your write-up should be clean.
  • Your proof must read coherently top-to-bottom.
  • Beginnings and ends of proofs must be clearly indicated. (For example, using Proof. . . \Box format.)
  • For each proof, state the claim you are proving.
    • This is not the same as rewriting the text of the problem. State only the claim you are actually
      proving. Sometimes this will be rather different from the text of the problem.
  • Use notation and terminology consistent with the textbook, course notes, or in-class usage. If you
    introduce new notation or terminology, you will need to clearly explain it; otherwise I will read your
    proof as though I have no idea what the notation or terminology mean.


For each letter grade, there are three kinds of specifications: exam performance, participation, and homework.

exam performance:

A: 85% of non-proof points on one exam; 70% of non-proof points on another exam; 70% of non-proof points on the final; 6 proofs on exams
B: 50% of non-proof points on one exam; 70% of non-proof points on another exam; 70% of non-proof points on the final; 4 proofs on exams
C: 50% of non-proof points on one exam; 50% of non-proof points on the final; 2 proofs on exams


A: complete all but one preticipation assignment and all but one reading and reflection assignment
B: complete all but two preticipation assignments and all but one reading and reflection assignment
C: complete all but three preticipation assignments and all but two reading and reflection assignments


A: 80 S-marked proofs, demonstrating 30 techniques, including 15 level-II techniques; compute three integrals
B: 65 S-marked proofs, demonstrating 20 techniques, including 10 level-II techniques; compute two integrals
C: 50 S-marked proofs, demonstrating 10 techniques; compute an integral

You will submit your S-graded proofs which meet the 'techniques' specifications as part of your Final Portfolio.  You can submit up to two each of each technique marked *.

level II I
basics of \mathbb{R} use Cauchy's Criterion

use the Monotone Sequence Property

use the Bolzano-Weierstraß Property

use the Nested Intervals Theorem

use an Archimedean principle

use the \epsilon-characterization of supremum or infimum

limits use the sequential characterization

compute a limit superior or limit inferior

use the \epsilon-\delta definition
continuity use the topological characterization

use Bolzano's Intermediate Value Theorem

use the \epsilon-\delta definition

use the sequential characterization

topology use the definition of open set

use the definition of closed set

use the sequential characterization of closed set

use the sequential characterization of open set

compactness use topological compactness

establish topological compactness

use sequential compactness

establish sequential compactness

use the Heine-Borel property for a set

establish the Heine-Borel property for a set

named theorems prove an ``at infinity" version of a Named Theorem *

prove an ``infinite limit" version of a Named Theorem *

derivatives use the Mean Value Theorem

use Taylor's Theorem

use Darboux's Theorem

use the limit definition of the derivative

use Carathéodory's definition of the derivative

integration integrate by showing a 'bad' set is small  use the Cauchy Criterion for integrability
convergence of functions establish uniform convergence

prove a sequence converges pointwise but not uniformly

series use the Root Test prove the series version of a Named Theorem about sequences *
sequential criteria formulate and prove a sequential criterion *


Explanation Points. Particularly cogent proof-writing may earn you an Explanation Point. You can redeem three Explanation Points for a Level-II Technique. Otherwise Explanation Points will contribute to +/- letter grades.