MAT 215 Final

MAT 215 Final - Final — Math 215 Fall 2008 You may use...

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Unformatted text preview: Final — Math 215 Fall 2008 You may use Rudin for the exam. In your answers, you may cite results proven in Rudin. No other source may be consulted. You may not discuss the exam with anyone. Remember to sign the pledge. The exam must be returned by noon on Friday January 16th to my mailbox in the department office on the 3 rd floor of Fine Hall. 24 hours, 8 questions I. Let R be the real numbers with the standard Euclidean metric. Let ( M, d ) be a metric space, and let f : R → M be a function satisfying | x − y | ≤ d ( f ( x ) , f ( y )) (1) for all x, y ∈ R . Let Im( f ) ⊂ M be the image of f . (i) Must f be 1-to-1 onto Im( f )? (ii) Must f be continuous? (iii) If f is a continuous function that satisfies (1), must Im( f ) ⊂ M be a closed subset? Recall the image denotes the set Im( f ) = { m ∈ M | ∃ x ∈ R such that m = f ( x ) } . Answer all questions with proof or counterexample. 1 2 II. Let √ 2 = 1 . 4142135 . . . ∈ R be the positive square root of 2. Con- sider the harmonic sequence a 1 = 1 , a 2 = 1 2 , a 3 = 1 3 , . . . (2) with a k = 1 k . Answer the following questions with proof.....
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This note was uploaded on 05/03/2010 for the course MAT 215 at Princeton.

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MAT 215 Final - Final — Math 215 Fall 2008 You may use...

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