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Definition of supremum

  1. Sep 2, 2010 #1
    If you're given a sequence [itex]\{x_n\}[/itex], do you have

    [tex]
    \sup_n x_n = \lim_{n\to \infty} \left( \max\limits_{1 \leq k \leq n} x_k \right)
    [/tex]

    I've never seen this definition before, but it makes sense.

    ...and if it's NOT the same as the supremum...what *is* it?
     
    Last edited: Sep 2, 2010
  2. jcsd
  3. Sep 2, 2010 #2
    It is the same as the supremum. But you should try to prove they're equal for yourself; it follows somewhat easily from the definitions, where sup is the least upper bound.
     
    Last edited: Sep 2, 2010
  4. Sep 2, 2010 #3

    HallsofIvy

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    Notice that this gives the supremum for a sequence- a countably infinite set. The supremum is defined for any set (assuming [itex]+\infty[/itex] is a valid supremum), countable or not.
     
  5. Sep 4, 2010 #4

    Landau

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    @HallsofIvy: I don't really see the relevance of your remark, but if you want to generalize: the supremum is defined for any partially ordered set.
     
  6. Sep 4, 2010 #5
    I think the relevance of his remark is this: It's not possible for the condition I listed to be the definition of the supremum of a set of real numbers because it doesn't even make sense for an uncountably infinite set; e.g., [itex][0,1][/itex]. But I think it works just fine for any countably infinite set.
     
  7. Sep 4, 2010 #6

    Landau

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    Sure it is not valid for subsets of R, in the same way it is not valid for arbitrary posets. But you were explicitly talking about sequences of reals, so...
     
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