Complete Notes.pdf - CHAPTER 3 THE REAL NUMBERS PRELIMINARIES Absolute Value Def Let x ∈ R The absolute value of x denoted | x | is |x| = x −x if x

Complete Notes.pdf - CHAPTER 3 THE REAL NUMBERS...

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CHAPTER 3: THE REAL NUM- BERS PRELIMINARIES: Absolute Value Def. Let x R . The absolute value of x , denoted | x | , is: | x | = x, if x 0 - x, if x < 0 Subscribe to view the full document.

Properties of absolute value: Let x, y R and let a 0. Then (a) | x | ≥ 0. (b) | x | ≤ a iff - a x a . (c) | xy | = | x | | y | . (d) | x + y | ≤ | x | + | y | . (Triangle inequality) Section 3.3. Boundedness and the Completeness Axiom There is a one-to-one correspondence between the set R of real numbers and the points P on the number line. There are two basic types of real numbers: 1. The rational numbers Q , and 2. the irrational numbers R - Q . Subscribe to view the full document.

Def. Let S R , S 6 = Ø. S is bounded above if there exists a number m R such that s m for all s S . m is called an upper bound for S . S is bounded below if there exists a number k R such that s k for all s S . k is a lower bound for S . S is bounded if it is bounded above and below.  Subscribe to view the full document.  Subscribe to view the full document.  Subscribe to view the full document.  Subscribe to view the full document.  Subscribe to view the full document.  Subscribe to view the full document.  Subscribe to view the full document.  Subscribe to view the full document.

The Completeness Axiom: Let S R , S negationslash = Ø. If S is bounded above, then S has a least upper bound. If S is bounded below, then S has a greatest lower bound. The real number system { R , + , · , < } is a complete ordered field . 41  • Fall '08
• Staff
• Topology, Mathematical analysis, Metric space, Limit of a sequence

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