Proving $\| x\| -\|Bx\| \leq \| x-Bx\|$ when $\|B\| \leq 1$

Question

If $B \in \mathbb R^{n\times n}$ and $x \in \mathbb R^n$, and $\|B\| \leq 1$, Can we prove the following statement? $$\|x\| -\|Bx\| \leq \|x-Bx\|$$

Any insight would be appreciated. Thank you.

Answer 1

For vectors $x, y$ in a normed vector Space we have $$ \lVert x \rVert - \lVert y \rVert \leq \lVert x - y \rVert $$

as a direct consequence of the triangle inequality. No information about $B$ is needed.

Answer 2

This follows from the reverse triangle inequality.

Answer 3

$$\left\| x\right\|=\left\| x-Bx+Bx\right\|\leq\left\| x-Bx\right\|+\left\|Bx\right\| $$