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:
 * $\forall x, y, z \in X: \big\lvert {\norm {x - z} - \norm {y - z}} \big\rvert \le \norm {x - y}$

Then:
 * $\forall x, y \in X: \big\lvert{\norm x - \norm y}\big\rvert = \big\lvert{\norm{x - 0} - \norm{y - 0} }\big\rvert \le \norm {x - y}$.