Comparing vector distance and norm values

Question

Given that $\mathrm{dist}(\mathbf u, \mathbf v) < R$, is it valid that $\|\mathbf v\| > \|\mathbf u\| - R$ in $\mathbb R^n$?

If so, how do you prove it? I'm aware that $\mathrm{dist}(\mathbf u, \mathbf v) = \|\mathbf u - \mathbf v\|$, but the triangle inequality doesn't seem to lead anywhere.

Answer 1

How about $$\|u\| = \| v + (u-v)\| \le \|v\| + \|u-v\| < \|v\| + R$$