Cauchy sequences is my proof correct?

[SMA_01]

Homework Statement

Let (x_n)_n$\in$ℕ and (y_n)_n$\in$ℕ be Cauchy sequences of real numbers.

Show, without using the Cauchy Criterion, that if z_n=x_n+y_n, then (z_n)_n$\in$ℕ is a Cauchy sequence of real numbers.

Homework Equations

The Attempt at a Solution

Here's my attempt at a proof:

Let (x_n) and (y_n) be Cauchy sequences. Let (z_n) be a sequence and let z_n=x_n+y_n.

Since (x_n) and (y_n) are Cauchy, $\exists$N$\in$ℕ such that,
|x_n-x_m|<ε/2, and
|y_n-y_m|<ε/2 for n,m≥N.

Let n,m≥N and let z_n,z_m$\in$(z_n).
Then,
|z_n-z_m|=|x_n-x_m|+|y_n-y_m|
<ε/2+ε/2=ε.
Therefore,
|z_n-z_m|<ε for all n,m≥N and hence, (z_n) is a Cauchy sequence of real numbers.

Is this correct?
Any input is appreciated.

Thanks.

 

[MathematicalPhysicist]

Your first step should be an inequality and not an equality, other than that seems ok, you just need to take the maximum of N1 and N2 where these indices are the ones for which

$$\forall n,m \geq N_1 \ |x_n-x_m|\leq \epsilon/2$$
$$\forall n,m \geq N_2 \ |y_n-y_m| \leq \epsilon /2$$