Introduction to Real Analysis, Fourth Edition [Wiley India Edition]

This text provides the fundamental concepts and techniques of real analysis for students in all of these areas. It helps one develop the ability to think deductively, analyses mathematical situations and extend ideas to a new context. Like the first three editions, this edition maintains the same spirit and user-friendly approach with addition examples and expansion on logical operations and set theory. There is also content revision in the following areas, introducing point-set topology before discussing continuity, including a more thorough discussion of limsup and limimf, covering series directly following sequences, adding coverage of lebesgue integral and the construction of the reals and drawing student attention to possible applications wherever possible. Chapter 1 preliminaries sets and functions. Mathematical induction. Finite and infinite sets. Chapter 2 the real numbers the algebraic and order properties of r. Absolute value and the real line. The completeness property of r. Applications of the supremum property. Intervals. Chapter 3 sequences and series. Sequences and their limits. Limit theorems. Monotone sequences. Subsequences and the bolzano-weierstrass theorem. The cauchy criterion. Properly divergent sequences. Introduction to infinite series. Chapter 4 limits limits of functions. Limit theorems. Some extensions of the limit concept. Chapter 5 continuous functions continuous functions. Combinations of continuous functions. Continuous functions on intervals. Uniform continuity. Continuity and gauges. Monotone and inverse functions chapter 6 differentiation the derivative. The mean value theorem. L'hospital's rules. Taylor's theorem. Chapter 7 the riemann integral riemann integral. Riemann integrable functions. The fundamental theorem. The darboux integral. Approximate integration. Chapter 8 sequences of functions pointwise and uniform convergence.