hw4 - MATH 410: Homework 4 Due in class Friday 2/24/2012...

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Unformatted text preview: MATH 410: Homework 4 Due in class Friday 2/24/2012 Section 2.4 1. For each of the following statements determine whether it is true or false and justify informally: (a) Every sequence in (0 , 1) has a convergent subsequence. * (b) Every sequence in (0 , 1) has a convergent subsequence that converges in (0 , 1). * (c) Every sequence of rational numbers has a convergent subsequence. * (d) If a sequence of nonnegative numbers converges, its limit is also nonnegative. * (e) Every sequence of nonegative numbers has a convergent subsequence. * Note: This is book problem 2. 2. Identify the first five terms of what appears to be a monotonically increasing subsequence of { cos( n ) } . Give the explicit n values and their cosines. * Note: This is not a book problem. 3. Show if { a n } is monotone and has a convergent subsequence then { a n } converges. ** Note: This is book problem 8. Section 3.1 1. Define f ( x ) = braceleftBigg x 2 if x x + 1 if x > (a) Show that f ( x ) is not continuous at...
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