Proving $\mid \parallel x \parallel - \parallel y \parallel \mid$ $\leqq$ $\parallel x-y \parallel$?

Question

Let $V$ be a normed vector space and I need to prove the follwing inequality $\mid \parallel x \parallel - \parallel y \parallel \mid$ $\leqq$ $\parallel x-y \parallel$ containing the norm and the absolute value of the real numbers. However things just get twisted and I cannot see how to prove it.... Could anyone please tell me how to prove it?

Answer 1

$||x||=||x-y+y|| \le ||x-y||+||y||$, hence

$(1)$ $||x||-||y|| \le ||x-y||$.

In a similar way we get

$(2)$ $||y||-||x|| \le ||y-x||$.

Since $||y-x||=||x-y||$, $(1)$ and $(2)$ give the result.