Reverse Triangle Inequality/Normed Vector Space

Theorem
Let $\left({X, \left\lVert{\cdot}\right\rVert}\right)$ be a normed vector space.

Then:
 * $\forall x, y \in X: \left\lVert{x - y}\right\rVert \ge \big\lvert{\left\lVert{x}\right\rVert - \left\lVert{y}\right\rVert}\big\rvert$

Proof
Let $d$ denote the metric induced by $\left\lVert{\cdot}\right\rVert$, that is,
 * $d \left({x, y}\right) = \left\lVert{x - y}\right\rVert$

From Metric Induced by Norm is Metric we have that $d$ is indeed a metric.

Then, from the Reverse Triangle Inequality as applied to metric spaces:
 * $(1): \quad \forall x, y, z \in X: \left|{d \left({x, z}\right) - d \left({y, z}\right)}\right| \le d \left({x, y}\right)$

Let $z = 0$.

Then $(1)$ translates to:
 * $\forall x, y, z \in X: \big\lvert{\left\lVert{x - 0}\right\rVert - \left\lVert{y - 0}\right\rVert}\big\rvert \le \left\lVert{x - y}\right\rVert$

Hence the result.