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# hw10solutions - Math 521 Advanced Calculus I Instructor J...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: February 12, 2010 Assignment 10 1. Let { x n } be a bounded sequence, and for each n N let s n := sup { x k : k n } and t n := inf { x k : k n } . Prove that { s n } and { t n } are monotone and convergent. Also prove that if lim s n = lim t n , then { x n } is convergent. [One calls lim s n the limit superior of { x n } , lim sup x n , and lim t n the limit inferior of { x n } , lim inf x n .] Since { x n } is bounded, we know that s n and t n exist for each n . Moreover, we have that X n +1 := { x k : k n + 1 } ⊂ { x k : k n } =: X n . Thus any upper bound for X n is automatically an upper bound for X n +1 . In particular s n is an upper bound for the set X n +1 . And since s n +1 is its least upper bound, it follows that s n +1 s n . Similarly, a lower bound for X n , namely t n , is automatically a lower bound for X n +1 . Since

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hw10solutions - Math 521 Advanced Calculus I Instructor J...

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