hw4 - ∃ K ∈ N such that ∀ n ≥ K x n ≤ x and x-x...

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MATH 444: ELEMENTARY REAL ANALYSIS HOMEWORK 4 Due date: Sep 24 (Wed) Exercises from the textbook. 3.1: 3 (b) and (d); 5 (b) and (d); 10; 12 3.2: 1 (b) and (d); 7 Out-of-textbook exercises. 1. Let ( x n ) be a sequence and x R . For each of the following conditions, determine whether it implies that x is the limit of ( x n ); if YES, then prove it, and if NO, give a counter-example (i.e. give an example of ( x n ) and x that satisfy the condition but ( x n ) does not converge to x ). (a) ε > 0 N N such that | x - x N | < ε . (b) ε > 0 K N such that n K , x - x n < ε . (c)
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Unformatted text preview: ∃ K ∈ N such that ∀ n ≥ K , x n ≤ x and x-x n < ε . (d) x = sup n ∈ N x n or x = inf n ∈ N x n . (e) x = sup n ∈ N x n and x = inf n ∈ N x n . (f) ∀ K ∈ N ∃ ε > 0 such that ∀ n ≥ K , | x n-x | < ε . (g) ∀ ε > ∃ K ∈ N such that ∃ n ≥ K , x n ∈ V ε ( x ). (h) ∀ ε > ∃ K ∈ N such that x-ε < inf n ∈ N x n ≤ sup n ∈ N x n < x + ε . (i) ∃ ε > ∃ K ∈ N such that ∀ n ≥ K , x n ∈ V ε ( x )....
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