So in fact X sup S Corollary 12 For a bounded sequence of real numbers s n the

So in fact x sup s corollary 12 for a bounded

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So, in fact, X = sup S. Corollary 1.2. For a bounded sequence of real numbers { s n } , the limit inferior is equal to the infimum of all subsequential limits of { s n } . Proof: The slick proof is the following. (1) The supremum of a set S of real numbers is equal to - inf {- x : x S } . (2) Consequently, show that lim n →∞ inf k n s k = - lim n →∞ sup k n - s k . (3) By the theorem, this is equal to - sup { subsequential limits of {- s n }} , which by step 1 is equal to inf { subsequential limits of { s n }} . Here is a more direct proof. Let > 0 . Let x = lim inf s n . By definition of limit, there exists N N such that when n N , then | x - x n | < 2 . Note that x n s k for all k n by definition of infimum. So, by Theorem 17.4, x n L for any subsequential limit L. Consequently, x x n + L + for any subsequential limit L. This is true for any > 0 , which shows that x L. Now, the subsequential limit L was arbitrary, which shows that x is a lower bound for S. Consequently, by definition of infimum, (1.3) x inf S. On the other hand, by definition of supremum, x n + 2 is not a lower bound for { s k } k n , so there exists s n k with n k n such that x n s n k x n + 2 . By the triangle inequality, | s n k - x | ≤ | s n k - x n | + | x n - x | ≤ for n k n N. Let k = 2 - k . Then, for each k, there exists s n k such that | s n k - x | ≤ k , n k < n k +1 . Thus, we have constructed a subsequence which converges to x. Hence, x S so inf S x. Together with (1.3), this shows that inf S = x.
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MATH 117 LECTURE NOTES FEBRUARY 24, 2009 3 Remark: Within this proof, we have shown Corollary 19.12, and in particular, given a bounded sequence { s n } , there exists a subsequence whose limit is lim inf s n and a subsequence whose limit is lim sup s n .
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  • Fall '08
  • Akhmedov,A
  • Real Numbers, Limits, Supremum, Limit of a sequence, Limit superior and limit inferior, Xn, subsequence, DR. JULIE ROWLETT

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