difference of Cauchy sequences is Cauchy

Question

Let $X$ be a normed space and let $(x_n)$ and $(y_n)$ be Cauchy sequences on $X$. Show that $z_n=\mid\mid x_n-y_n\mid\mid$ is also a Cauchy sequence on $\mathbb{R}.$

My answer:

$ | z_n-z_m | = | \ \ ||x_n-y_n||-||x_m-y_m|| \ \ | \leq ||x_n-y_n||+||x_m-y_m||$

i stuck at this step. I need hint.

Answer 1

I think you don't need to mix the x's and y's:

$$||z_n-z_m||=||x_n-y_n-x_m+y_m||\le||x_n-x_m||+||y_m-y_n||\le\ldots$$