Foundations of Analysis

Course title

33005900

Middle-level Diploma Policy (mDP)

Program / Major	mDP	Goals
IoT Course	DP-4a・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Software Course	DP-4b・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Media Course	DP-4c・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Data Science Course	DP-4d・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Mechatronics Course	DP-4・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Architecture and Architectural Engineering Course	DP-4a・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Environmental Systems and Urban Planning Course	DP-4b・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Bioscience Course	DP-4a・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Biomedical Engineering Course	DP-4b・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Sports Engineering Course	DP-4c・3	専門分野と他分野を関連付ける素養主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Mathematical Sciences Course	DP-4・2	キャリアを見据えた高度な専門知識現象の背後にある数理構造やデータのパターンを理論的に解析し、社会や自然科学、工学における複雑な課題に対して数理的視点から解決戦略を提案できる。

Purpose of class

Analysis is the discipline that deals with limits. For example, in mathematics up to high school level, convergence was simply defined as “approaching infinitely close,” but what “infinitely close” means mathematically was unclear. In university-level mathematics, this is precisely defined using inequalities. This is the so-called “ε-N argument” or “ε-δ argument.” By rigorously defining convergence, it becomes possible to discuss convergence not just in terms of getting close, but also in terms of the rate of convergence. This allows for the complete proof of properties that could not be rigorously proven in high school mathematics. In this course, we will learn how limits are discussed in analysis and study its important theorems.

Course description

In the first half, we learn the ε-N method through “convergence of sequences,” followed by the ε-δ method through “continuous functions.” It is particularly important to note that both methods are clever devices for expressing limits as inequalities. In the latter half, we define “supremum and infimum” for sets where an “order relation” is established. We focus on discussing the “completeness of real numbers” within the set of all real numbers, which is a representative example of such sets. Furthermore, we explore the characteristics of “uniform continuity” and “uniform convergence,” which are crucial concepts in analysis.

Goals and objectives

To be able to use ε-N arguments to discuss the convergence of sequences.
To be able to explain the completeness of real numbers to others.
To be able to use ε-δ arguments to discuss the convergence of functions.

Relationship between 'Goals and Objectives' and 'Course Outcomes'

	Mid-term exam	Final exam	Total.
1.	25%		25%
2.	25%	25%	50%
3.		25%	25%
Total.	50%	50%	-

Evaluation method and criteria

Mid-term exam and Final exam. When the minimum and maximum scores of the midterm and final examinations are m and M, respectively, a student is considered to have passed if m≥30 and M≥60, where M is the grade.

Language

Japanese

Class schedule

	Class schedule	HW assignments (Including preparation and review of the class.)	Amount of Time Required
1.	Convergence of sequences 1 (ε-N method, etc.)	Regardless of whether you do your prep work, make sure you review.	190minutes
2.	Convergence of sequences 2 (properties of convergent sequences, etc.)	Regardless of whether you do your prep work, make sure you review.	190minutes
3.	Cauchy sequence	Regardless of whether you do your prep work, make sure you review.	190minutes
4.	Continuous functions 1 (ε-δ argument, etc.)	Regardless of whether you do your prep work, make sure you review.	190minutes
5.	Continuous functions 2 (basic properties, etc.)	Regardless of whether you do your prep work, make sure you review.	190minutes
6.	Midterm exam and feedback	Review lecture notes and ensure you can solve the textbook’s example problems and exercises.	190minutes
7.	Order relationship 1 (upper limit, lower limit, etc.)	Regardless of whether you do your prep work, make sure you review.	190minutes
8.	Order relations 2 (cutting, etc.)	Regardless of whether you do your prep work, make sure you review.	190minutes
9.	Completeness of real numbers 1 (convergence of bounded monotonic sequences, interval contraction theorem, etc.)	Regardless of whether you do your prep work, make sure you review.	190minutes
10.	Completeness of real numbers 2 (Bolzano-Weierstrass theorem, intermediate value theorem, maximum and minimum theorem, etc.)	Regardless of whether you do your prep work, make sure you review.	190minutes
11.	Uniform continuity	Regardless of whether you do your prep work, make sure you review.	190minutes
12.	Riemann integral	Regardless of whether you do your prep work, make sure you review.	190minutes
13.	Uniform convergence	Regardless of whether you do your prep work, make sure you review.	190minutes
14.	Final exam and feedback	Review lecture notes and ensure you can solve the textbook’s example problems and exercises.	190minutes
Total.	-	-	2660minutes

Feedback on exams, assignments, etc.

ways of feedback	specific contents about "Other"
Feedback outside of the class (ScombZ, mail, etc.)

Textbooks and reference materials

Textbook: Yohei Komori, Sets and Topology (Nippon Hyoronsha) (same textbook as “Logic and Set Theory” in the spring semester). For reference books, please refer to the “Reference Books for Courses Taught” section on the laboratory website.
https://sites.google.com/shibaura-it.ac.jp/shingo/

Prerequisites

Review the spring semester courses “Logic and Set Theory” and “Calculus 1” beforehand.

Office hours and How to contact professors for questions

Every Wednesday during the academic term, after the fourth period (it is advisable to contact us in advance when visiting).

Regionally-oriented

Non-regionally-oriented course

Development of social and professional independence

Course that cultivates an ability for utilizing knowledge

Active-learning course

More than one class is interactive

Course by professor with work experience

Work experience	Work experience and relevance to the course content if applicable
N/A	N/A

Education related SDGs:the Sustainable Development Goals

4.QUALITY EDUCATION
9.INDUSTRY, INNOVATION AND INFRASTRUCTURE

Last modified : Fri Mar 27 04:02:45 JST 2026