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
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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)
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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 .
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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.
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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
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  • Fall '08
  • Staff
  • Topology, Mathematical analysis, Metric space, Limit of a sequence

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