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

[Aerostd]

Homework Statement

Let x_{n} = y_{n} + z_{n}
Also, x_{n}>0, y_{n}>0, z_{n}>0. We also know that x_{n} converges.

Prove that y_{n} converges.

Homework Equations

I want to use the Cauchy criterion because the limits are not given. So start with an \epsilon >0 [/itex]. Then there exists N such that for all n, m > N[/itex], <br /> <br /> | x_{n} - x_{m} | < \epsilon [/itex]<br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> <br /> Well, substituting the expression for x_{n},<br /> <br /> <br /> | y_{n} + z_{n} - y_{m} - z_{m} | = | y_{n} - y_{m} + z_{n} - z_{m} |<br /><br /> <br /> Here, I can't use the triangle inequality because it goes in the wrong direction. Basically I don't know if z_{n} - z_{m} is greater than zero or less than zero which is causing problems. Is there another method which I can use to prove this?

 

[micromass]

So you know nothing about z_n??

I think it's very easy to find a counterexample to this.

 

[Aerostd]

So you know nothing about z_n??

I think it's very easy to find a counterexample to this.

Actually I just thought of one. Thanks.

as always.