01 Logic and Sets in Mathematics
02 Properties of real numbers
03 Limit of a sequence of real numbers
04 Limits of real functions
05 Differentiation of real functions
06 Riemann integral of real functions
07 Infinite series of real numbers
08 Real analytical function

Analysis is the branch of mathematics that studies the properties of spaces that have both algebraic and topological structures, and the properties of functions defined on such spaces.
In particular, the undergraduate course in basic analysis defines limits based on a rigorous number system and explains the properties of space and functions, serving as a stepping stone to various fields of mathematics.

This book is a textbook for those studying the fundamentals of analysis, and is written at a level suitable for those who have taken at least one semester each of undergraduate-level set theory and calculus.
The content is structured primarily for those who are studying hermeneutics for the first time, but includes a variety of examples and expanded content to ensure that it is also useful for those with experience studying hermeneutics.

Before presenting the theorem, this book explains why it is necessary and presents a method of thinking for its proof.
And I explained how the contents of each chapter are connected to each other, and organized it so that the reader can understand the whole organically and comprehensively, rather than feeling that the contents are separate.

By introducing the concepts of open and closed sets in developing the content and consistently applying a topological perspective throughout the entire unit, the reader's thinking can be easily expanded beyond functions defined in Euclidean space to functions defined in general metric spaces.
Applying a topological perspective may seem difficult at first, but if you understand it properly, it will be of great help in studying subsequent subjects such as complex analysis, differential geometry, topology, and functional analysis.

Sections marked with an asterisk (*) in the table of contents do not have content connected to subsequent units, so those who are new to hermeneutics can skip these units.
The theorems in the text are marked with an asterisk because they are too difficult for beginners to understand, so beginners can skip over them.
The proofs marked with an asterisk are too difficult for beginners to understand, so beginners can just check the contents of the theorem and skip the proof.

Practice problems are presented by level, so readers can choose and solve problems that are appropriate for their level.
For those who are studying hermeneutics for the first time, it is recommended to study the examples and examples in the text carefully and focus on problems that help them understand the concepts.
If you can easily solve problems that require you to understand the concept or have studied hermeneutics at least once, it would be a good idea to focus your study on problems that require you to apply the concept.
Anyone who wants to improve their skills should try solving the skill building problems.
The Leap Problems are comprised of a variety of problems related to subsequent subjects, the history of mathematics, and mathematics education, and provide guidance for truly deep understanding and study of analysis.

This book has been revised several times since its first edition was published in 2005.
The second edition, released in June 2005, added content on distance space, and the third edition, released in January 2008, adopted a structure that took learners' psychology into account.
In the third revised edition released in January 2010, the editorial design was improved with readability in mind.

Here are the changes in this 4th edition:
Added a section on 'Logic and Sets in Mathematics'.
We have provided learning objectives for each unit to guide the direction of learning.
The content has been adjusted to suit the level of an introduction to hermeneutics.
From the publication of the first edition to the publication of the fourth edition, many people have provided feedback during the process of planning and editing the content, helping to correct errors and create an even better book.
We have done our best to provide accurate and good content, but we believe there may still be some shortcomings, such as typos or logical errors.
If you find something that needs to be fixed or have suggestions for better content, please email designeralice@daum.net
Please send it to me.

You can get the latest information about this book by visiting the Alice in Mathland community at http://aliceinmathland.com.
We hope you'll visit our cafe and share information with other visitors, brainstorming and solving various problems together.
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If you study hard and share it with others, it will come back to you multiplied.

I hope that people studying mathematics can get a lot of information online.
This book, along with many other books and materials I've published online, are a reflection of my desire to create such an Internet culture.
I hope this book will be helpful to you, the readers.
I would like to thank all of you who have shown interest in this book.