Norm and Cauchy sequence.

Question

If someone could please show me how to show that if $x_n$ is a Cauchy sequence then $x_n \over ||x_n||$ is a Cauchy sequence as well?

Thanks, I hope I've been clear enough.

Answer 1

Let our working domain be $\mathbb{R}$ and $(x_n)$ be a non-zero sequence of points of real numbers. Then First we have to note that $$y_n=\dfrac{x_n}{|x_n|}=\pm 1.$$ If $x_n>0,\forall n\in\mathbb{N}$ then $(y_n)=(1)$ is Cauchy.
Similar thing is valid for, if $x_n<0,\forall n\in\mathbb{N}.$

Conversely, suppose $(x_n)$ be any divergent sequence of positive reals (or divergent sequence of negative reals). Then although $(x_n)$ is not cauchy, $(y_n)$ is Cauchy.