Real Analysis and Infinity 1st Edition by H Sedaghat IISSBN ‎ 0192895621 ‎ 978-0192895622

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ISBN 10: ‎ 0192895621
ISBN 13: ‎ 978-0192895622
Author: H Sedaghat

Real Analysis and Infinity presents the essential topics for a first course in real analysis with an emphasis on the role of infinity in all of the fundamental concepts. After introducing sequences of numbers, it develops the set of real numbers in terms of Cauchy sequences of rational numbers, and uses this development to derive the important properties of real numbers like completeness. The book then develops the concepts of continuity, derivative, and integral, and presents the theory of infinite sequences and series of functions.

Topics discussed are wide-ranging and include the convergence of sequences, definition of limits and continuity via converging sequences, and the development of derivative. The proofs of the vast majority of theorems are presented and pedagogical considerations are given priority to help cement the reader’s knowledge.

Preliminary discussion of each major topic is supplemented with examples and diagrams, and historical asides. Examples follow most major results to improve comprehension, and exercises at the end of each chapter help with the refinement of proof and calculation skills.

Real Analysis and Infinity 1st Table of contents:

• Introduction: The Intersection of Real Analysis and Infinity

  • Overview of Real Analysis
  • The Role of Infinity in Mathematics
  • Key Concepts: Convergence, Limits, and the Infinite
  • A Historical Perspective on Infinity in Mathematics

• Chapter 1: The Foundations of Real Analysis

  • The Real Number System: Definitions and Properties
  • The Axioms of Real Numbers
  • Completeness of the Real Numbers
  • The Structure of the Real Line

• Chapter 2: Sequences and Series

  • Defining Sequences: Convergence and Divergence
  • Limit of a Sequence and its Behavior
  • Series and the Convergence Tests
  • Infinite Series and Their Applications

• Chapter 3: Functions and Continuity

  • The Concept of a Function in Real Analysis
  • Continuity: Definition and Properties
  • Intermediate Value Theorem and its Significance
  • Limits and Continuity at Infinity

• Chapter 4: Limits and Infinity

  • The Concept of Limits in Real Analysis
  • Limits at Infinity and Their Applications
  • Infinite Limits and Indeterminate Forms
  • One-Sided Limits and Infinite Sequences

• Chapter 5: Differentiation and the Infinite

  • Definition of the Derivative
  • The Mean Value Theorem and its Connection to Infinity
  • Behavior of Functions at Infinity: Limits of Derivatives
  • Applications of Differentiation to Infinite Series and Sequences

• Chapter 6: Integration and Infinite Processes

  • The Riemann Integral: Definition and Properties
  • Improper Integrals and Handling Infinite Limits
  • Infinite Sums and their Relation to the Integral
  • The Role of Infinity in Integrating Functions

• Chapter 7: Infinite Sets and Cardinality

  • The Concept of Infinite Sets
  • Cardinality and the Size of Infinite Sets
  • Countable vs. Uncountable Infinity
  • Cantor’s Diagonal Argument and the Continuum Hypothesis

• Chapter 8: The Paradoxes of Infinity

  • Zeno’s Paradoxes: A Philosophical Introduction
  • Infinite Sums and the Paradox of Convergence
  • Hilbert’s Hotel and Infinite Sets
  • The Concept of Actual vs. Potential Infinity

• Chapter 9: Compactness, Convergence, and Infinity

  • Compactness and Its Role in Real Analysis
  • Heine-Borel Theorem: Compact Sets in the Real Line
  • Convergence in Infinite-Dimensional Spaces
  • The Compactness of Infinite Sets in Real Analysis

• Chapter 10: Metric Spaces and Infinity

  • Introduction to Metric Spaces
  • The Concept of Distance and Infinity in Metric Spaces
  • Convergence and Compactness in Infinite Metric Spaces
  • Topological Properties of Infinite Sets

• Chapter 11: The Role of Infinity in Modern Real Analysis

  • Infinity and Its Impact on Contemporary Mathematical Theory
  • The Use of Infinity in Fractals and Chaos Theory
  • Infinity in Functional Analysis and Beyond
  • The Continuum Hypothesis and Set Theory in Real Analysis

• Chapter 12: Philosophical Perspectives on Infinity

  • Infinity in Philosophy: Historical Context
  • The Paradox of Actual Infinity vs. Potential Infinity
  • Infinite Processes in Philosophy and Physics
  • Infinity and the Infinite Regress in Logical Arguments

• Epilogue: The Infinite Journey of Real Analysis

  • Reflection on the Role of Infinity in Modern Mathematics
  • Challenges and Future Directions in Real Analysis
  • Infinity in Mathematics and Its Continuing Mysteries

• Appendices

  • A: Proofs of Key Theorems
  • B: Commonly Used Formulas and Concepts in Real Analysis
  • C: Further Reading on Infinity and Real Analysis

• References

  • List of Cited Works and Recommended Texts

• Index

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H Sedaghat,Real Analysis,Infinity