home4 - segments shown. 1-7 1-6 1-5 1-4 1-3 1-2 1 1 3....

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MATH 4317 HOMEWORK #4 DUE: July 10, 2007 Work the following problems and hand in your solutions. You may work together with other people in the class, but you must each write up your solutions independently. A subset of these will be selected for grading. Write LEGIBLY on the FRONT side of the page only, and STAPLE your pages together. 1. Prove that if A R p , then its boundary ∂A is a closed subset of R p . Note: This is NOT a one-line proof! 2. Let E be the union of the countably many line segments in R 2 shown below, AND the vertical line segment L = { (0 , y ) : 0 y 1 } . Prove that E is connected. The surprise here is that E is not polygonally path-connected! (You don’t have to prove that.) Note: E does NOT include any points on the x -axis other than the endpoints of the line
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Unformatted text preview: segments shown. 1-7 1-6 1-5 1-4 1-3 1-2 1 1 3. Problem 14 I. Let X = ( x n ) be a sequence of strictly positive real numbers such that lim ( x n +1 /x n ) < 1. Show that for some r with 0 < r < 1 and some C > 0, then we have < x n < Cr n for all suciently large n N . Use this to show that lim ( x n ) = 0. 4. Let ( x n ) be a sequence of numbers in R p . Suppose that there is an x R p such that every subsequence ( y n ) of ( x n ) has a subsequence ( z n ) of ( y n ) such that lim z n = x . Show that lim ( x n ) = x . Hint: Proof by contradiction. What does it mean to say that x n does not converge to x ? 1...
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