Introduction to Analysis I

Academic unit or major

Undergraduate major in Mathematics

Instructor(s)

Yoshiyuki Kagei / Masaharu Tanabe

Class Format

Lecture/Exercise (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-8 Mon

Class

-

Course Code

MTH.C201

Number of credits

110

Course offered

2025

Offered quarter

1Q

Syllabus updated

Mar 19, 2025

Language

Japanese

Syllabus

Course overview and goals

In this course we give a rigorous formalization of "limits" of sequences and functions by means of the "epsilon-delta" definitions.
We also learn the differential calculus rigorously, and in particular the Taylor expansion as an excellent polynomial approximation.
Each lecture will be followed by a recitation (a problem-solving session). This course will be succeeded by "Introduction to Analysis II" in the second quarter.

The students will learn how to write the mathematical analysis logically. The students will become familiar with the "epsilon-delta" definitions and proofs, and be able to describe limits of real numbers rigorously.

Course description and aims

At the end of this course, students are expected to:
-- Understand the construction of irrational numbers
-- Be familiar with limits superior and limits inferior
-- Be able to state and prove propositions about limits of sequences and functions by using "epsilons and deltas".
-- Understand some important properties of continuous functions, such as the intermediate value theorem, and the existence of the maximum/minimum.
-- Be able to calculate polynomial approximations of a given function by using the Taylor expansion or asymptotic expansion.

Keywords

continuity of real numbers, infimum, supremum, limit superior, limit inferior, Cauchy sequence, continuous function, derivative, Taylor expansion

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

This is a standard lecture course with recitation sessions.
Homework will be assigned every week. There will be occasional quizzes.

Course schedule/Objectives

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

None required

Reference books, course materials, etc.

None required

Evaluation methods and criteria

Based on the attendance, quizzes, reports, and the problem solving situation in the recitation sessions. Details will be provided in the class.

Related courses

• MTH.C202 ： Introduction to Analysis II
• MTH.C203 ： Introduction to Analysis III
• MTH.C204 ： Introduction to Analysis IV

Prerequisites

Students are expected to have passed Calculus I/II, Linear Algebra I/II, and their reciｔations.