An Introduction to Real Analysis

Ravi P. Agarwal, Cristina Flaut, Donal O'Regan

• Verlag: Chapman and Hall/CRC
• Jahr: 2018
• ISBN:
  9781351180627
• Auflage: 1
• Thema: Mathématiques
• Thema: Mathématiques générales

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.

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...