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Cauchy sequences

  1. Oct 25, 2004 #1
    Let {an}(n goes from 1 to infinity) be a sequence. For each n define:
    sn=Summation(j=1 to n) of aj
    tn=Summation(j=1 to n) of the absolute value of aj.

    Prove that if
    {tn}(n goes from 1 to infinity)
    is a Cauchy sequence, then so is
    {sn}(n goes from 1 to infinity).


    I started this proof with the definition of a Cauchy sequence. Pick an N large enough so that n,m>N makes
    |an - am| < epsolon.
    So if tn is Cauchy, we have
    |tn-tm| < epsolon.
    tn-tm = summation|an|-summation|am| = |an|+|an-1|+...+|am+1|
    so now
    |an| + |an-1| +...+ |am+1| < epsolon
    but
    |an + an-1 + ... + am+1| < |an|+|an-1|+...+|am+1|
    by triangle inequality.
    so now
    |an + an-1 +...+ am+1| < epsolon
    but
    |an + an-1 + ... + am+1| = |sn - sm|
    so now
    |sn-sm| < epsolon, and therefore Cauchy.


    Can anybody tell me if this makes sence? Or at least tell me how to write out "summation from n=1 to infinity" on here in symbols??? Thanks so much!
     
  2. jcsd
  3. Oct 25, 2004 #2

    NateTG

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    Science Advisor
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    There's a thread on [tex]\LaTeX[/tex] somewhere around here...
    [tex]s_n=\sum_{i=0}^{n}a_i[/tex]

    The proof looks ok too.
     
  4. Oct 25, 2004 #3
    Thank you SOOO much!
     
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