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If xn = yn + zn, and xn, yn, zn >0, and xn xonverges, then yn converges.

  1. Dec 14, 2011 #1
    1. The problem statement, all variables and given/known data
    Let [itex]x_{n} = y_{n} + z_{n}[/itex]
    Also, [itex] x_{n}>0 [/itex], [itex] y_{n}>0 [/itex], [itex] z_{n}>0 [/itex]. We also know that [itex] x_{n} converges.[/itex]

    Prove that [itex] y_{n} [/itex] converges.


    2. Relevant equations

    I want to use the Cauchy criterion because the limits are not given. So start with an [tex] \epsilon >0 [/itex]. Then there exists [itex] N [/itex] such that for all [tex] n, m > N[/itex],

    [tex] | x_{n} - x_{m} | < \epsilon [/itex]


    3. The attempt at a solution


    Well, substituting the expression for [itex] x_{n}[/itex],

    [itex]
    | y_{n} + z_{n} - y_{m} - z_{m} | = | y_{n} - y_{m} + z_{n} - z_{m} |
    [/itex]

    Here, I can't use the triangle inequality because it goes in the wrong direction. Basically I don't know if [itex] z_{n} - z_{m} [/itex] is greater than zero or less than zero which is causing problems. Is there another method which I can use to prove this?
     
  2. jcsd
  3. Dec 14, 2011 #2

    micromass

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    So you know nothing about [itex]z_n[/itex]??

    I think it's very easy to find a counterexample to this.
     
  4. Dec 14, 2011 #3
    Actually I just thought of one. Thanks.

    :confused: as always.
     
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