Chapter 2

प्रस्तावना

The Preface introduces the textbook 'वास्तविक विश्लेषण एवं दूरीक समष्टि' (Real Analysis and Metric Spaces), published by Vardhman Mahaveer Open University, Kota. This book is designed according to the curriculum for B.Sc. Part II Mathematics, Paper I. The authors have aimed to make the language simple, engaging, and accessible, using appropriate English terminology alongside Hindi explanations. The content is structured to facilitate both conceptual understanding and competitive exam preparation. The Preface acknowledges the contributions of various scholars and references standard texts to ensure factual accuracy. The book is intended to guide students through the study of real analysis and metric spaces, emphasizing clarity and depth.

• Textbook is tailored for B.Sc. Part II Mathematics, Paper I.
• Language is simple and accessible, with English terms used alongside Hindi.
• Content is based on standard references for factual accuracy.
• Aims to help students in both academic and competitive exam contexts.
• Acknowledges contributions from expert authors.
• 📌 वास्तविक विश्लेषण (Real Analysis): The study of real numbers and their properties.
• 📌 दूरीक समष्टि (Metric Spaces): Mathematical structures where distances between elements are defined.

इकाई 1 : वास्तविक संख्या निकाय (Real Numbers System) इकाई की रूपरेखा

This section outlines the structure of Unit 1, which focuses on the Real Number System. It lists the topics to be covered, including objectives, introduction, field axioms, order axioms, boundedness, completeness, properties of real numbers, and exercises. The unit is organized to systematically build understanding from basic definitions to advanced properties, such as the Archimedean property and density property. Each sub-section is designed to introduce and explain key concepts, definitions, and theorems relevant to real numbers.

• Unit 1 covers the Real Number System in detail.
• Topics include field axioms, order axioms, boundedness, completeness, and properties.
• Sub-sections address specific properties like Archimedean and density.
• Exercises and self-assessment questions are included.
• 📌 Field: A set with addition and multiplication satisfying certain axioms.
• 📌 Order: Arrangement of numbers based on size.
• 📌 Boundedness: Concept of upper and lower bounds for sets.