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Homework Help: Test Review 4 - is it this easy?

  1. Oct 15, 2005 #1
    Let (sn) be a real sequence and s in R. Prove that lim sup (sn) < s implies
    sn < s for n large.

    My answer seems too easy. Is there anything missing?

    Given lim sup (sn) < s.
    By definition of lim sup, we know lim N->infinity sup {sn: n > N} < s
    Then for n > N, we must have sn < s.

  2. jcsd
  3. Oct 15, 2005 #2


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    What exactly is the "definition of lim sup"? How does
    lim N->infinity sup {sn: n > N} < s follow from it?
  4. Oct 15, 2005 #3
    The definition in our books is that:

    lim sup {sn: n>N}

    i.e. For large n, the lim sup sn is the limit of all of the suprema. They also defined lim sup sn to be exactly the supremum of the set of subsequential limits.
  5. Oct 15, 2005 #4
    Also, the fact that lim sup sn < s for some real number s is given.

    Then I just made the substitution lim sup sn = lim n->infinity sup{sn: n>N}
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