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

  • Thread starter Aerostd
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  • #1
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Homework Statement


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.


Homework 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]


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?
 

Answers and Replies

  • #2
22,089
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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.
 
  • #3
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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.
Actually I just thought of one. Thanks.

:confused: as always.
 

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