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|&lt;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 &gt; 0 there is a natural number N such that if n, m &gt; N then |(x_n + y_n) - (x_m + y_m)| &lt; \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.