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 [tex] \epsilon >0 [/itex]. Then there exists $N$ 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 $x_{n}$,

$| y_{n} + z_{n} - y_{m} - z_{m} | = | y_{n} - y_{m} + z_{n} - z_{m} |$

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.