TITLE: How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?
QUESTION [7 upvotes]: I'm trying to show that $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$. A hint would be nice.

REPLY [9 votes]: Observe that 
$\lVert x \rVert  = \lVert (x -y) +y \rVert \leq \lVert (x -y) \rVert + \lVert y \rVert$
which gives 
$\lVert x \rVert - \lVert y \rVert \leq \lVert x -y \rVert$ ... $(1)$
Further,
$-(\lVert x \rVert - \lVert y \rVert ) \leq \lVert (y -x) \rVert = \lVert (x -y) \rVert $... $(2)$
From $(1)$ and $(2)$ result follows.