Is xn+yn a Cauchy sequence if xn and yn are Cauchy sequences?

[kathrynag]

Homework Statement

Let xn and yn be Cauchy sequences.
Give a direct argument that xn+yn is a Cauchy sequence that does not use the Cauchy Criterion or the Algebraic Limit Theorem.

Homework Equations

The Attempt at a Solution

given epsilon>0 there exists an N in the natural numbers such that whenever m,n>N, it follows that:
$$\left|xn-xm\right|<epsilon$$ and $$\left|yn-ym\right|$$<epsilon
I'm not sure where to go next.

 

[JG89]

To show that $$x_n + y_n$$ is Cauchy, you want to prove that for any $$\epsilon > 0$$ there is a natural number $$N$$ such that if $$n, m > N$$ then $$|(x_n + y_n) - (x_m + y_m)| < \epsilon$$, right?

Note that $$|(x_n + y_n) - (x_m + y_m)| = |(x_n - x_m) + (y_n - y_m)|$$. Now use the triangle inequality and remember that both sequences $$x_n, y_n$$ are Cauchy.