Norm of $p$-adic number well defined.

Question

After the construction of the $p$-adic numbers as equivalent Cauchy sequences of rational numbers with respect to the $p$-adic absolute value, we define the norm of $\lambda\in\mathbb{Q}_p$ to be $|\lambda|_p=\operatorname{lim}_{n\rightarrow\infty}|x_n|_p$, where $(x_n)$ is any Cauchy sequence representing $\lambda$. But how do we know the limit exists? Can we say that from knowing $(x_n)$ is Cauchy that $|x_n|$ is Cauchy (and so converges since $\mathbb{R}$ is complete?) If so, why is this true?

Answer 1

Use reverse triangle law.

$$ | |x_n|_p - |x_m|_p | \leq |x_n - x_m|_p$$

So $|x_n|_p$ is Cauchy.