Math 112: Introductory Real Analysis (Spring 2025)
See
Canvas course website for the most up-to-date information.
Meeting time and location:
Mon Wed 10:30 AM - 11:45 AM
at
Science Center 507.
Instructor:
Sunghyuk Park
Email: sunghyukpark at math dot harvard dot edu
Office hours:
Mon 11:50 AM - 12:50 PM at
Science Center 231
Course assistants:
CA list and resources
| Course Assistant |
Email |
Office Hours (Recitations) |
| Maya Robinson |
mayarobinson at college dot harvard dot edu |
Mon 8:00 - 9:00 PM
at Leverett Dining Hall |
| Gili Zaid |
gilizaid at college dot harvard dot edu |
Tue 3:00 - 4:15 PM
at Leverett Dining Hall |
| Autumn Shin |
autumnshin at college dot harvard dot edu |
Wed 7:30 - 9:00 PM
at Quincy Dining Hall |
| Karen Song |
karensong at college dot harvard dot edu |
Thu 4:30 - 5:30 PM
at Adams Dining Hall |
| Trace Baxley |
tbaxley at college dot harvard dot edu |
Fri 10:30 - 11:30 AM
at Eliot Dining Hall |
| Marcello Laurel |
marcellolaurel at college dot harvard dot edu |
Sun 8:00 - 9:00 PM
at Lowell Dining Hall |
Course goals:
This course is an introduction to mathematical analysis and the theory behind calculus, with an emphasis on learning to understand and construct proofs.
It will cover limits and continuity in metric spaces, uniform convergence and spaces of functions, and the Riemann integral.
More specifically, here are the topics I am planning to talk about:
- Main topics: (the first 7 chapters of Rudin)
- the real and complex number systems
- basic topology
- numerical sequences and series
- continuity, differentiation
- the Riemann-Stieltjes integral
- sequences and series of functions
Tentative schedule
| Date |
Topic |
Date |
Topic |
| 1/27 |
The Real and Complex Number Systems; ordered sets, least-upper-bound property |
1/29 |
the real field $\mathbb{R}$, and consequences of its l.u.b. property |
| 2/3 |
the complex field, Euclidean spaces |
2/5 |
Basic Topology; cardinality of sets, Cantor's diagonal argument, uncountability of $\mathbb{R}$ |
| 2/10 |
metric spaces, open sets, topological spaces |
2/12 |
closed sets, limit points, interior, closure |
| 2/17 |
No class (President's Day) |
2/19 |
compact sets |
| 2/24 |
Heine-Borel theorem, perfect sets |
2/26 |
Cantor set, connected sets |
| 3/3 |
Numerical Sequences and Series; convergent sequences |
3/5 |
subsequences, Cauchy sequences |
| 3/10 |
Midterm Week!
upper and lower limits |
3/12 |
series |
| 3/17 |
No class (Spring Recess) |
3/19 |
No class (Spring Recess) |
| 3/24 |
the root and ratio tests |
3/26 |
summation by parts, rearrangements |
| 3/31 |
Continuity; limits of functions, continuous functions |
4/2 |
continuity and compactness |
| 4/7 |
continuity and connectedness, discontinuity |
4/9 |
Differentiation; the derivative of a real function, mean value theorems |
| 4/14 |
l'Hospital's rule, Taylor's theorem |
4/16 |
The Riemann-Stieltjes Integral; definition and existence of the integral |
| 4/21 |
properties of the integral |
4/23 |
integration and differentiation |
| 4/28 |
Sequences and Series of Functions; uniform convergence |
4/30 |
Stone-Weierstrass theorem |
| 5/5 |
Finals Week! |
|
|
Textbook:
Rudin - Principles of Mathematical Analysis [
pdf]
Prerequisites:
Math 19a/b or 21a/b and an ability to write proofs or concurrent enrollment in Mathematics 101 (or an equivalent background in mathematics).
Office hours / recitations:
There will be weekly office hours / recitations.
You are all encouraged to attend, whether you have a question or not.
I will answer any questions about the class and discuss more topics in real analysis following any request of the students attending.
Grading:
Homework will be assigned weekly, every Wednesday and has to be submitted the following Wednesday by midnight.
The solutions can either be scanned or typed and uploaded on Canvas.
Homework will count for at least 70% of the final grade.
Late homework is not accepted, unless under special circumstances.
Collaborative work on homework is accepted but you must write your own solution as well as the names of the collaborators and any other reference used (books, MathOverflow, ...), see below for Harvard College Honor Code.
There will be a midterm (take-home) and a final exam (take-home), which will count for 30% of the grade.
Special rule about the use of AI (such as ChatGPT): you are not allowed to use AI tools to solve any questions (homework, exam, midterm), nor ask it to suggest resources to solve these questions. Doing so is a violation of Harvard Honor Code.
Midterm and final exam:
The midterm will be given on
Mar 10, 2025 (Mon) and the final exam will be given on
May 5, 2025 (Mon).
You will have one week to return each of them.
Harvard College Honor Code:
"Members of the Harvard College community commit themselves to producing academic work of integrity –
that is, work that adheres to the scholarly and intellectual standards of accurate attribution of sources, appropriate collection and use of data, and transparent acknowledgement of the contribution of others to their ideas, discoveries, interpretations, and conclusions.
Cheating on exams or problem sets, plagiarizing or misrepresenting the ideas or language of someone else as one's own, falsifying data, or any other instance of academic dishonesty violates the standards of our community, as well as the standards of the wider world of learning and affairs."