Jacob Bernstein





Math 405: Introduction to Real Analysis

Course Description

This is an introduction to real analysis. Topics covered in the course will include, The Logic of Mathematical Proofs, Construction and Topology of the Real Line, Continuous Functions, Differential Calculus, Integral Calculus, Sequences and Series of Functions. There will be 10 problem sets (20% of final grade), two in class midterm exams (20% each) and one final exam (40%).

Lectures are Monday and Wednesday 1:30-2:45 in Hodson 315. Section meets Friday 1:30-2:20 in Hodson 315.

Problem sets will be due in class on Wednesdays (see below for dates). No late homework will be accepted. The lowest grade will be dropped.

Lecturer: Jacob Bernstein. Lecturer Office hours: Monday, 3-4pm and Tuesday 10-11am or by appointment in Krieger 408.

TA: Letian Chen. TA Office hours: Tuesday, 3-5pm in Krieger 211.


The course text is
  • R. Strichartz, “The Way of Analysis," Rev. Ed. (Available on Amazon for ~$40)
A supplementary text (dual licensed under Creative Commons Attribution-Noncommercial-Share Alike 4.0 License and Creative Commons Attribution-Share Alike 4.0 License) is


There will be three exams. Two in class midterms and a final.

The dates of the exames are
First Midterm: Monday, October 7th.
Second Midterm: Monday, November 11th.
Final Exam: Wednesday, December 11th, 9am-12pm.

(Tentative) Schedule

Week 0 and 1 (8/29 & 9/4) : Logic of Mathematical Proofs

Read JL: 0.3; See also Sections 1-4 of M. Taylor's notes. RS: 1.1, 1.2, 1.3, 1.4.
If you need a more in depth introduction to basic logic and proofs see this book.
No homework due.

Week 2 (9/9 & 9/11): Construction of the Real Numbers

Read JL: 1.1 and 1.2. See RS: 2.1, 2.2;
Problem Set 1 due. Solutions to selected problems.

Week 3 (9/16 & 9/18): Construction of the Real Numbers (cont.); Topology of the Real Number Line

Read JL: 1.3, 1.4 and 1.5. See RS: 2.3
Problem Set 2 due. Solutions to selected problems.

Week 4 (9/23 & 9/25): Topology of the Real Number Line (cont.)

Read JL: 2.1, 2.2. See RS: 3.1, 3.2, 3.3
Problem Set 3 due. Solutions

Week 5 (9/30 & 10/2): Topology of the Real Line (cont.)

Read JL: 2.3, 2.4 and 2.5
Lecture held in Krieger 413 on Wednesday! Section held in Ames 234 on Friday!
Problem Set 4 due. Solutions

Week 6 (10/7 & 10/9): First Midterm; Continuous Functions

Read JL: 3.1, 3.2. See RS: 4,1 4.2
No homework due.
Practice Midterm (Solutions). More Practice. (Solutions), Practice. (Solutions). The exam will not cover properties of continuous functions.
First Midterm Solutions

Week 7 (10/14 & 10/16): Continuous Functions (cont.)

Read: JL: 3.2, 3.3, 3.4. See RS: 5.1, 5.2
Problem Set 5 due. Solutions

Week 8 (10/21 & 10/23): Differential Calculus

Read: JL: 4.1, 4.2, 4.3. See RS: 5.3, 5.4, 6.1
Problem Set 6 due. Solutions

Week 9 (10/28 & 10/30): Integral Calculus

Read JL: 5.1, 5.2. See RS: 6.1
Problem Set 7 due. Solutions

Week 10 (11/4 & 11/6): Integral Calculus (cont.)

Read JL: 5.3, 5.4, 5.5. See RS: 6.2
Problem Set 8 due. Solutions

Week 11 (11/11 & 11/13): Second Midterm; Sequences and Series of Functions

Read JL: 6.1. See RS: 7.2, 7.3
No homework due.
Practice Midterm (Solutions). More Practice (Solutions). Even more Practice (Solutions). More Practice
Second Midterm Solutions

Week 12 (11/18 & 11/20): Sequences and Series of Functions (cont.)

Read JL: 6.2, 2.6. See RS: 7.3, 7.4
Problem Set 9 due. Solutions

Week 13: Thanksgiving Break

No homework due.

Week 14: (12/2 & 12/4): Picard iteration and the existence theory for ODEs.

Read JL: 6.3.
Problem Set 10 due. Solutions

Final (12/11)

I will have office hours by appointment.
Practice Final (with solutions). More practice. Solutions.
Even more practice (with solutions).
Solutions to Final.

Students with disabilities

Students with documented disabilities or other special needs who require accommodation must register with Student Disability Services. After that, remind the instructor of the specific needs at least two weeks prior to each exam; the instructor must be provided with the official letter stating all the needs from Student Disability Services.

JHU ethics statement

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of students and/or the chairman of the Ethics Board beforehand. Read the "Statement on Ethics" at the Ethics Board website for more information.

If a student is found responsible through the Office of Student Conduct for academic dishonesty on a graded item in this course, the student will receive a score of zero for that assignment, and the final grade for the course will be further reduced by one letter grade.

Fall 2019 -- Department of Mathematics, Johns Hopkins University.