Reverse triangle inequality

Proposition (Reverse triangle inequality). In a normed vector space $(X,\|\cdot\|)$, for all $x,y\in X$,

Context. This inequality is a key tool for showing that norm convergence implies convergence of norms and for proving continuity of the norm mapping.

Proof sketch. From the triangle inequality,

Swap $x$ and $y$ to get $\|y\|-\|x\|\le \|x-y\|$, and combine both.