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Intro to Analysis (Cauchy)

  1. Mar 5, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove the following assertion: Suppose {xn} and {yn} are Cauchy sequences of real number. If {xn} is a cauchy sequence and for every η>0 there exists a pos. int. N such that for every n>N so that abs(xn-yn)<η then {yn} is a Cauchy sequence.


    2. Relevant equations
    None



    3. The attempt at a solution
    We will prove that {yn} is a cauchy sequence by showing that for every ε>0 there exists a pos. int. N so that both n and m >N so that abs(yn-ym)<ε.

    Consider ε>0 arbitrary.
    Since {xn} is a cauchy sequence and by hypothesis for every η>0 there exists an N so that every n>N abs(xn-yn)<η. Choose such an N.
    Consider n,m>N arbitrary.

    Then I know I need to get from abs(yn-ym) to ε but unsure how to use what I have to get there correctly and if I have the rest of the proof right.
     
  2. jcsd
  3. Mar 6, 2012 #2
    Use the Cauchy sequence {xn} as your bridge, think about triangle inequality, also note that η may be set according to ε
     
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