midterm - Modern Analysis 1 Midterm Answer at most five...

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Unformatted text preview: Modern Analysis 1 Midterm Answer at most five questions. 1. Let S be a complete ordered set. For each λ ∈ Λ let Aλ ⊂ S be nonempty and let A = ∪λ∈Λ Aλ be bounded above; let α = sup A and for each λ ∈ Λ let αλ = sup Aλ . Prove or disprove: α = sup{αλ : λ ∈ Λ}. 2. (i) State the Cauchy-Schwarz inequality, with a precise necessary and sufficient condition for equality. N 2 (ii) Find the minimum value of n=1 an given that the N real numbers a1 , . . . , aN have sum equal to 1. 3. Let K = {1/n : Z n > 0} ∪ {0} with the usual (Euclidean) metric. (i) Prove directly that K is compact. (ii) Which subsets of K are compact? Prove. (iii) Which subsets of K are precompact? Explain. 4. Let A and B be disjoint, nonempty compact subsets of a metric space. Define δ = inf {d(a, b) : a ∈ A, b ∈ B }. Prove that there exist a0 ∈ A and b0 ∈ B such that δ = d(a0 , b0 ). 5. (i) Prove that a real sequence that is increasing and bounded above converges. √ (ii) Prove that the real sequence (an )∞ defined by a0 = 2 and n=0 √ n 0 ⇒ an+1 = 2 + an converges and find its limit. 6. Let (an )n∈N be a sequence of positive reals with the property that an → 0 as n → ∞. Prove that there exists a permutation π of N such that the rearranged sequence (aπ(n) )n∈N is decreasing: aπ(0) aπ(1) .... Decide (with justification) what happens to aπ(n) as n → ∞. 1 ...
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This note was uploaded on 11/09/2011 for the course MAA 5228 taught by Professor Robinson during the Fall '09 term at University of Florida.

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