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Real analysis problem

  1. Oct 12, 2008 #1
    1. The problem statement, all variables and given/known data
    Let E be nonempty subset of R which is bounded above (thus, a = sup E exists)
    Does there exist a strictly monotone sequence in E which converges to a?


    2. Relevant equations



    3. The attempt at a solution
    I've been thinking about just taking a monotone bounded (this must be true by condition of E) sequence of rationals in E(an interval on the real line) which converges to the supremum (endpoint of the interval). I'm not sure how to formally construct this.
     
  2. jcsd
  3. Oct 12, 2008 #2

    Dick

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    Is E really an interval on the real line? I don't think so. Otherwise it's too easy. If you know nothing about E, then what do you do if E is finite? If E is infinite then just pick any element of E to start the sequence. How can you pick the next element?
     
  4. Oct 27, 2008 #3
    anybody can help me ??
    how to solve this problem;

    show dat the sequence xn = (n-10)/(n+10) is both bounded & monotone
     
  5. Oct 27, 2008 #4

    statdad

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    Let [tex] a [/tex] denote the sup.

    As has been arleady noted, if the set [tex] A [/tex] is finite it is easy. Suppose [tex] A [/tex] is not finite.
    Choose

    [tex]
    \epsilon_1 = 1
    [/tex]

    Then there is an [tex] x_1 \in A [/tex] such that

    [tex]
    |x_1 - a| < \epsilon_1
    [/tex]

    Now let [tex] \epsilon_2 = 1/2 [/tex]

    There is an [tex] x_2 \in A [/tex] such that

    [tex]
    |x_2 - a| < \epsilon_2
    [/tex]

    Continue this process with [tex] \epsilon_3, \epsilon_4, \dots [/tex] to generate a sequence
    [tex] \{x_i\}_{i=1}^\infty [/tex]. This is the sequence you need.
     
  6. Oct 27, 2008 #5

    Dick

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    Write it as 1-20/(n+10). Next time start a new thread if you have a new question.
     
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