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Intro to Analysis in-class_Part_10

# Intro to Analysis in-class_Part_10 - Q this is equivalent...

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1.3 The rational numbers 25 Proof 1. Using the OBVIOUS note, -| a | ≤ a ≤ | a | -| b | ≤ b ≤ | b | - ( | a | + | b | ) a + b ≤ | a | + | b | | a + b | ≤ || a | + | b || = | a | + | b | . Proof 2. | a + b | 2 = ( a + b )( a + b ) = a 2 + 2 ab + b 2 a 2 + 2 | a || b | + b 2 = ( | a | + | b | ) 2 . This allows for quantitative version of “if x is close to y and y is close to z , then x is close to z ”: let | x - y | < ε and | y - z | < ε . Then: | x - z | = | x + ( - y + y ) - z | = | ( x - y ) + ( y - z ) | ≤ | x - y | + | y - z | < ε + ε = 2 ε. So if we want x to be within 1 2 of z , find x within 1 4 of y and z within 1 4 of y . Other forms of Δ ineq: | x - y | ≥ | x | - | y | | x - y | ≥ || x | - | y || fl fl fl fl fl n X i =1 x i fl fl fl fl fl n X i =1 | x i | Proof. Fun! (And required) Theorem 1.3.13 (Axiom of Archimedes) . Let x > 0 . Given any M (no matter how large), y Q such that xy > M . By the field properties of

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Unformatted text preview: Q , this is equivalent to: Let x > 0. Given any ε (no matter how small), ∃ y ∈ Q such that 0 < xy < ε . These are also true for R . A basic idea of analysis: a < b = ⇒ ∃ c ∈ ( a,b ) ∩ R . 26 Math 413 Preliminaries I.e., a < c < b and c ∈ R . Question 4. What does this mean? ∀ ε > , | a-b | < ε 1.4 Axiom of Choice Given a sequence of nonempty sets A 1 ,A 2 ,... , the product Q ∞ k =1 A k is nonempty....
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Intro to Analysis in-class_Part_10 - Q this is equivalent...

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