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

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Modern Analysis 1 Midterm Answer at most five questions. 1. Let S be a complete ordered set. For each λ Λ let A λ S be nonempty and let A = λ Λ A λ be bounded above; let α = sup A and for each λ Λ let α λ = sup A λ . Prove or disprove: α = sup { α λ : λ Λ } . 2. (i) State the Cauchy-Schwarz inequality, with a precise necessary and sufficient condition for equality. (ii) Find the minimum value of N n =1 a 2 n given that the N real numbers a 1 , . . . , a N have sum equal to 1. 3. Let K = { 1 /n : Z 3 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
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