- 277 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android
eBook - ePub
An Introduction to Real Analysis
About this book
This book provides a compact, but thorough, introduction to the subject of Real Analysis. It is intended for a senior undergraduate and for a beginning graduate one-semester course.
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Yes, you can access An Introduction to Real Analysis by Ravi P. Agarwal,Cristina Flaut,Donal O'Regan in PDF and/or ePUB format, as well as other popular books in Mathematics & Functional Analysis. We have over one million books available in our catalogue for you to explore.
Information
Chapter 1
Logic and Proof Techniques
We begin this chapter with the definition of mathematical statements, and introduce some logical connectives that we will use frequently in this book. We will also discuss some commonly used methods to prove mathematical results.
By a mathematical statement or proposition, we mean an unambiguous composition of words that is true or false. For example, two plus two is four is a true statement, and two plus three is seven is a false statement. However, x − y = y − x is not a proposition, because the symbols are not defined. If x − y = y − x for all x, y real numbers, then this is a false proposition; if x − y = y − x for some real numbers, then this is a true proposition. Help me please, and your place or mine—are also not statements. A single letter is always used to denote a statement. For example, the letter p may be used for the statement eleven is an even number. Thus, p:11 is an even number. A statement is said to have truth value T or F according as the statement is true or false. For example, the truth value of p:1 + 2 + … + 10 = 55 is T, whereas for p:12 + 22 + 32 = 15 is F. The knowledge of truth value of a statement enables us to replace it by some other “equivalent” statement. From given statements, new statements can be produced by using the following standard logical connectives:
1. Negation, ~:If p is a statement, then its negation ~ p is the statement not p. Thetruth valueof ~ p is F or T according as the truth value of p is T or F. Thus, if p: seven is even number, then ~ p: seven is not an even number, or seven is an odd number.
2. Implication, ⇒: If from a statement p another statement q follows, we say p implies q and write p ⇒ q. The truth value of p ⇒ q is F only when p has truth value T and q has the truth value F. For example, x = 7 ⇒ x2 = 49. If n is an even integer, then n + 1 is an odd integer.
3. Conjuction, ∧: The statement p and q is denoted as p ∧ q and is called the conjunction of the statements p a...
Table of contents
- Cover
- Half Title
- Title Page
- Copyright Page
- Dedication
- Table of Contents
- Preface
- 1 Logic and Proof Techniques
- 2 Sets and Functions
- 3 Real Numbers
- 4 Open and Closed Sets
- 5 Cardinality
- 6 Real-Valued Functions
- 7 Real Sequences
- 8 Real Sequences (Contd.)
- 9 Infinite Series
- 10 Infinite Series (Contd.)
- 11 Limits of Functions
- 12 Continuous Functions
- 13 Discontinuous Functions
- 14 Uniform and Absolute Continuities and Functions of Bounded Variation
- 15 Differentiable Functions
- 16 Higher Order Differentiable Functions
- 17 Convex Functions
- 18 Indeterminate Forms
- 19 Riemann Integration
- 20 Properties of the Riemann Integral
- 21 Improper Integrals
- 22 Riemann-Lebesgue Theorem
- 23 Riemann-Stieltjes Integral
- 24 Sequences of Functions
- 25 Sequences of Functions (Contd.)
- 26 Series of Functions
- 27 Power and Taylor Series
- 28 Power and Taylor Series (Contd.)
- 29 Metric Spaces
- 30 Metric Spaces (Contd.)
- Bibliography
- Index