Vector inequalities with known bounds

Question

Let us assume that we have two vectors $\mathbb{v}$ and $\mathbb{u}$ such that

$\|\mathbb{v} + \mathbb{u}\| \leq A$

and

$\|\mathbb{u}\| \leq B$

where $A > B > 0$

The question is:

is $\|\mathbb{v}\| \leq A - B$ valid?

Answer 1

By the reverse triangle inequality, we have $\|v\|-\|u\|\leq \|v+u\|\leq A$ and hence $\|v\|\leq A+B$. I'm afraid we can't do better than that with the information given, unless I've made a gross error.

Answer 2

Actually it's not.

Suppose that $\|\mathbb{v}\| \leq A-B$ is true. Then

$\|\mathbb{v} + \mathbb{u} - \mathbb{u}\| \leq A-B$

which, by the reverse triangle inequality becomes

$\bigg| \|\mathbb{v} + \mathbb{u}\| - \|\mathbb{u}\| \bigg| \leq \|(\mathbb{v} + \mathbb{u}) - \mathbb{u}\| \leq A-B$

which means that

$\|\mathbb{v} + \mathbb{u}\| - \|\mathbb{u}\| \leq A -B$

and hence, by the assumption that $\|\mathbb{v} + \mathbb{u}\| \leq A$

$- \|\mathbb{u}\| \leq -B \Leftrightarrow \|\mathbb{u}\| \geq B$

which is a contradiction.