Simple vector inequality

Question

I was doing a long proof just now that had assumed that

$||x|| - ||y|| \le ||x+y||$

I thought that I had done this before, but what I had done was actually unrelated:

$||x-y|| \le ||x||+||y||$

Could I have a hint towards the first one?

Answer 1

Here's a hint: $x = (x + y) - y$.

Answer 2

Both the inequalities are equivalent. To get the first one from the second, since you seem to have proved the second, set $x-y = x_1$. This implies that $x = x_1 + y$. Plug this into the second inequality you have to get $$\lVert x_1 \rVert \leq \lVert x_1 + y \rVert + \lVert y\rVert$$ Hence, $$ \lVert x_1 \rVert - \lVert y \rVert \leq \lVert x_1 + y \rVert$$