Math 425a: Fundamental Concepts of Analysis
USC Fall 2020

Instructor: Kyler Siegel (
Lectures: 11:00 - 11:50am (LA time), Mondays, Wednesdays, and Fridays via Zoom

Teaching assistant: Linfeng Li (
Discussions sessions: 9:00 - 10:50am (LA time), Tuesdays via Zoom


This is an introductory course in real analysis. After covering the foundations of the real numbers and (naive) set theory, we will discuss metric spaces, elementary topology, calculus, and function space theory. While many of these subjects, for example the real numbers or calculus, will be quite familiar to you, we will take a much more rigorous approach than you've likely seen before, proving all important results and not taking anything for granted. As you will see, our human intuition can sometimes lead us astray, and many pathologies and exotica arise if we are not extremely careful with our assumptions.

For many students this may be their first course in rigorous pure mathematics. As such, this is a challenging course despite the seemingly simple subject material. We will put particular emphasis on learning how to invent and write careful mathematical proofs.


We will assume familiarity with calculus, especially limits, continuity, integration, differentiation, the fundamental theorem of calculus, and so on. We will also assume some familiarity with the basic structure of mathematical proofs, especially proof by induction, reductio ad absurdum, etc.


Some of the topics we will cover over the course of the semester (time permitting) include:

  • foundations of the real numbers (Dedekind cuts, the least upper bound property, completeness)
  • cardinality (naive set theory, countable versus uncountable sets, uncountability of the real numbers)
  • metric spaces (open and closed sets, closure and interior, continuity)
  • Cantor sets
  • toolkit of counterexamples (topologist's sine curve, Devil's staircase, etc)
  • compactness (sequential and in terms of coverings)
  • topological spaces (continuity, convergence, connectedness)
  • differentiation (mean value theorem, L'Hospital's rule, Taylor approximations)
  • the Riemann-Stieltjes integral (integrable functions, the fundamental theorem of calculus)
  • series (absolute versus conditional convergence, tests for convergence)
  • function spaces (time permitting)


  • Primary: Real Mathematical Analysis, 2nd edition - Charles Chapman Pugh.
  • Secondary: Principles of Mathematical Analysis, 3rd edition - Walter Rudin

    Note: we will be roughly following the first three chapters of Pugh, and parts of the fourth chapter, time permitting. I strongly recommend that you the relevant sections in advance so that you can get more out of the lectures. You might find it helpful to reread some of the sections several times, and to try to prove some of the theorems on your own before reading the proof. If you have additional time, as an additional reference I recommend the well-known book by Rudin.

    Structure of the course.

    The material of this course will roughly follow the first three chapters of the textbook by Pugh (including some but not all of the more advanced subsections), plus parts of the fourth chapter (time permitting). In addition to the material covered in lecture, students will be expected to follow along with the material in the textbook, ideally in advance of the lectures.

    Live lecture notes and recordings.

    The lectures and discussion sessions will be held via Zoom. The links and passwords can be found on Blackboard. Afterwards, Zoom recordings will appear on Blackboard. I strongly urge all students to attend the live lectures and discusssions if at all possible. Moreover, you are encouraged to actively participate by following along and asking and answering questions. We prefer that you turn on your video feed (unless bandwidth issues arise) in order to create a more responsive and social atmosphere.

    We understand that in some cases certain students will not be able to attend all live lectures due to time zone difficulties or other extenuating circumstances. In this case you should let the instructor know beforehand, and it is your responsibility to follow along with the recordings and other class materials.

    The lectures will primarily involve notes written in real time on an ipad. In addition to viewing lectures in Zoom via screen sharing, you will have the option of viewing the lecture notes in real time via this link. Some of you might find this useful if you wish to backreference something that happened earlier in the lecture. The completed past lecture notes will also be stored there. Note: the syncing is not instanteous, so you may sometimes need to refresh the document.

    Communication and discussion boards.

    USC has opted to use the messaging application Slack to facilitate communication between amongst students, and between students, instuctors, and TAs. Details about our Slack workspace will follow. Note: if there is interest we could also create a Piazza discussion board for this class.

    Problem sets.

    In addition to the lectures, there will be weekly problem sets. These will be listed on this website (see the schedule below) and will be handed in via Gradescope. Late homeworks will not be accepted. However, we will drop the lowest problem set score.

    Much of the course material will be developed in the problem sets. It is very important to do all of the problem sets to the best of your ability and to challenge yourself to solve the problems on your own, as this is the most effective way to absorb the material. We expect students to devote a significant amount of time to the problem sets.

    While working on the problem sets, you are allowed to consult or collaborate with your peers, as well as textbooks and the internet (apart from cheating websites such as Chegg or Cramster). However, you must write down attributions for any peer, textbook, website etc from which you took any significant ideas. Moreover, you must attempt all problems on your own and your submitted solutions must be written out originally and individually. Submissions which are copied or suspiciously similar are subject to being rejected and potential disciplinary action.


    There will be two midterm exams (roughly one third and two thirds of the way through the semester) and one final exam, which will take place via Gradescope. The midterms will occur during usual class time, with accomodations for students in different time zones. See the schedule below for tentative dates. Please keep in mind that the first midterm occurs very early - don't be caught off guard!

    Grading scheme.

    Homeworks: 25%, midterm exams: 20% each, final exam: 35%. We will drop the lowest homework grade.

    Office hours.

    Kyler will hold weekly office hours via Zoom. During the first week of class, these will be held immediately after the lectures, or by email appointment. Subsequently, you will sign up for office hours via this signup sheet. We will use the same Zoom link as the main lecture for that day. The schedule will be updated every one or two weeks. You should sign up at least 15 minutes (and preferably one hour) beforehand to make sure that Kyler shows up. Note that it is perfectly fine for several people to sign up for the same time slot, or you can just show up to any time slot where someone has already signed up. On the other hand, if you would like to have private office hours (e.g. to discuss a sensitive matter) please indicate so in the comment when you sign up. Also feel free to write in the comments what you would like to discuss, which might be helpful for other students with similar interests. If you would like to attend office hours with Kyler but cannot make any of these times, please don't hesitate to send Kyler an email and he will try his best to accommodate. Note: please do not send substantive math questions to Kyler via email.

    Kyler's weekly office hours are currently:

  • Mondays 12-2pm
  • Wednesdays 12-1pm

    Linfeng's office hours will be held via the Math Center. You are also encouraged to seek guidance in the Math Center even if you cannot attend Linfeng's office hours.

    Tentative schedule

    Note: this schedule is tentative and will be continuously updated to adapt to the pace of the course. Please check back regualrly for updates and problem set assignments.
    Date Material References Problem Set
    Monday 8/17/20 Introduction and course logistics. Three disturbing examples: space-filling curves, the devil's staircase, and the Riemann rearrangement theorem. Pugh: preface, §1.1 Problem set 1 (due Wednesday 8/26/20 by 11:59pm LA time).
    Wednesday 8/19/20 Naive set theory. Sets, maps, injections, surjections, equivalence relations. § 1.1
    Friday 8/21/20 More set theory. Unions, intersections, differences, partitions. Bijections and invertible maps. Russel's paradox. § 1.1
    Monday 8/24/20 Dedekind cuts and construction of the real numbers. § 1.2 Problem set 2 (due Wednesday 9/2/20 by 11:59pm LA time).
    Wednesday 8/26/20 Properties of the real numbers. Proof of the least upper bound property. § 1.2
    Friday 8/28/20 More on the real numbers. Cauchy sequences and completeness of the real numbers. § 1.2
    Monday 8/31/20 More on Cauchy sequences and completeness of the real numbers. § 1.2, §1.3
    Wednesday 9/2/20 Euclidean space and inner product spaces. §1.2, §1.3 Problem set 3 (due Wednesday 9/9/20 by 11:59pm LA time).
    Friday 9/4/20 Cardinality and uncountability of the real numbers. §1.4
    Monday 9/7/20 (labor day - NO CLASS)
    Wednesday 9/9/20 More on cardinality. §1.4, §1.5 Problem set 4 (due Wednesday 9/16/20 by 11:59pm LA time).
    Friday 9/11/20 More on cardinality. §1.5
    Monday 9/14/20 Introduction to metric spaces. §2.1
    Wednesday 9/16/20 Midterm 1 (no class). See Blackboard announcement and fake midterm on Gradescope for details. Fake midterm with exam instructions. Midterm 1 solutions.
    Friday 9/18/20 Metric space basics. Continuity and sequential continuity. §2.1, §2.2 Problem set 5 (due Wednesday 9/23/20 by 11:59pm LA time).
    Monday 9/21/20 Open and closed sets. §2.3
    Wednesday 9/23/20 More on open and closed sets. Isometries and homeomorphisms. §2.3 Problem set 6 (due Wednesday 9/30/20 by 11:59pm LA time). Problem 2(b) updated 9/25/20.
    Friday 9/25/20 Topology and topological continuity. Homeomorphisms and isometries. §2.3
    Monday 9/28/20 Connectedness and completeness. §2.3,§2.5
    Wednesday 9/30/20 Compactness §2.4 Problem set 7 (due Wednesday 10/7/20 by 11:59pm LA time).
    Friday 10/2/20 More on compactness, complete proof of Heine-Borel theorem. §2.4
    Monday 10/5/20 More on compactness and connectedness. §2.4, §2.5
    Wednesday 10/7/20 More on connectedness. Generalized intermediate value theorem. §2.5 Problem set 8 (due Wednesday 10/14/20 by 11:59pm LA time).
    Friday 10/9/20 More on connectedness. Path connectedness. Covering compactness. §2.5, §2.7
    Monday 10/12/20 The Cantor set. §2.8
    Wednesday 10/14/20 More on Cantor sets. §2.8 Problem set 9 (due Wednesday 10/28/20 by 11:59pm LA time).
    Friday 10/16/20 More on Cantor sets. Further metric space concepts.
    Monday 10/19/20 Further metric space concepts.
    Wednesday 10/21/20 Midterm 2 (no class). Topics list. Midterm 2 solutions.
    Friday 10/23/20 Differentiability. Review of key facts from calculus. §3.1 Problem set 10 (due Wednesday 11/4/20 by 11:59pm LA time).
    Monday 10/26/20 More on differentiability basics (mean value theorem, L'Hospital's rule, etc). §3.1
    Wednesday 10/28/20 More on differentiability. Lipshitz funtions. Pathologies. §3.1
    Friday 10/30/20 More on differentiability pathologies. Intermediate value theorem for derivatives. Continuous differentiability. Taylor approximation. §3.1
    Monday 11/2/20 Riemann integral: introduction and basics. §3.2
    Wednesday 11/4/20 More on the Riemann integral. Darboux integrability. §3.2 Problem set 11 (due Friday 11/13/20 by 11:59pm LA time).
    Friday 11/6/20 More properties of the Riemann integral and characterization of Riemann integrable functions. Uniformly continuous functions are Riemann integrable. §3.2
    Monday 11/9/20 Monotone functions are Riemann integrable. The Riemann-Lebesgue theorem. §3.2
    Wednesday 11/11/20 More on the Riemann-Lebesgue theorem. §3.2
    Friday 11/13/20 Antiderivatives and the fundamental theorem of calculus. §3.2 Final exam topics list.
    Wednesday 11/18/20 Final exam 11am-1pm LA time via Zoom. (Alternative session: 9pm-11pm LA time, must arrange beforehand.)