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Convergent sequence

  1. Sep 30, 2009 #1
    For a sequence in the reals

    {an} converges to a, show {|an|} converges to |a|.

    For any e>0 the exists an N s.t. for any n>N |an-a|<e

    I want to use this inequality, but there is something funny going on. I do not know how to justify it.

    |an-a|[tex]\leq[/tex]||an|-|a||
     
  2. jcsd
  3. Sep 30, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Look at three separate cases.

    1) a> 0. Can you show that, for some N, for all n> N [itex]a_n> 0[/itex]? (Take [itex]\epsilon= a/2[/itex].)

    2) a< 0. Can you show that, for some N, for all n> N [itex]a_n< 0[/itex]?

    3) a= 0. Here, [itex]||a_n|- a|= ||a_n||= |a_n|.
     
  4. Sep 30, 2009 #3
    Okay I see how to break it down case wise and find N accordingly. That will work nicely.

    However, I was hoping to use the reverse triangle inequality but I run into the double abs. value. It just doesn't look right to say that
    for any e>0 there exists and N s.t. for any n >N

    |an - a| < e
    and
    |an - a| [tex]\geq[/tex] |an| - |a|
    implies

    e>||an| - |a||

    but if I showed this wouldn't it be true?
     
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