Sequences of Real Numbers

Sequences of Real Numbers is a Khmer-language mathematical analysis guide written for Grade 11 students, university students and independent learners. It introduces the fundamental concepts used to define, classify and analyse real-number sequences. The book covers subsequences, monotonic sequences, bounded sequences, convergent sequences, Cauchy sequences and adjacent sequences. Each topic is supported by definitions, mathematical examples, worked solutions, theorems and practice exercises that help readers develop rigorous reasoning and problem-solving skills.

This publication introduces the study of real-number sequences as an important foundation of mathematical analysis. It was prepared as a learning and self-study resource for Grade 11 students, university students and readers who want to strengthen their understanding of sequences, limits and convergence.

The book begins with the definition of a real-number sequence and then develops the ideas of subsequences and monotonic behaviour. It explains how to determine whether a sequence is increasing, decreasing or constant, and how to establish upper and lower bounds.

Later sections examine convergent sequences through formal limit definitions, monotone convergence results and Cauchy sequences. The final part introduces adjacent sequences and their shared limit. Definitions and theorems are accompanied by worked examples and practice exercises to help readers connect mathematical theory with problem-solving techniques.

Readers are encouraged to study each definition carefully, follow the reasoning in the worked solutions and attempt the exercises independently.

Sequences of Real Numbers is the third mathematics publication I have compiled. I prepared it as a supporting resource for Grade 11 students, university students and independent learners who want to develop a stronger understanding of mathematical analysis.

The book introduces key concepts related to real-number sequences through definitions, theorems, examples, worked solutions and practice exercises. I have tried to organise the material in a clear and useful form, but the publication may still contain errors or areas that can be improved.

I welcome constructive feedback from teachers, students and readers. I hope this book will support learners in developing mathematical reasoning, studying independently and building a stronger foundation for more advanced mathematics.

I would like to express my sincere appreciation to the teachers, students, fellow mathematics learners and readers who have encouraged my continued work in mathematics education.

I am grateful to everyone who has provided feedback, asked questions and supported the preparation of my mathematics publications. Their encouragement has helped me continue improving these learning materials.

I hope this book will be a useful resource for learners studying sequences, limits and introductory mathematical analysis.