I was asked to proof that if the sequence $\{x_{n}\}_{n=1}^{\infty}$ is a Cauchy sequence in a normed space, then the sequence of norms $\{\Vert x \Vert\}_{n=1}^{\infty}$ converges.
I started by using the subtraction version of the triangle inequality and assuming $\Vert . \Vert_{1}$ as the corresponding norm.
$$\vert \Vert x_{n} \Vert - \Vert x_{m} \Vert \vert \leq \Vert x_{n} - x_{m} \Vert$$
But I only concluded with the sequence of norms being Cauchy, which doesn't necessarily implies it converges.
$$\Vert x_{n} - x_{m} \Vert < \epsilon$$
$$\vert \Vert x_{n} \Vert - \Vert x_{m} \Vert \vert \leq \Vert x_{n} - x_{m} \Vert < \epsilon$$
$$\vert \Vert x_{n} \Vert - \Vert x_{m} \vert \vert < \epsilon$$
From here I just got stuck, and don't know how to proceed. I was also thinking of assuming that, since the norm is in either $\mathbb{C}$ or $\mathbb{R}$ (both complete spaces), then the sequence should converge.
Thanks...