Vector inequality for a scalar difference of two vectors in $\mathbb{R}^n$.

A student posed an interesting problem to me the other day and embarrassingly I could not prove or disprove it even though it appears relatively simple.

The question was:

Given vectors $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ with $\|\mathbf{v}\|$,$\|\mathbf{w}\|\leq R$ and distinct scalars $a, b > 0$ does there exist a constant $C=C(R)$ (depending only on $R$, not on $a$ or $b$) such that

$$\|a\mathbf{v}-b\mathbf{w}\|\leq C |a-b|\\$$

The proof for the one-dimensional case was simple but in higher dimensions it doesn't seem as obvious to prove.

Any help in proving/disproving this would be greatly appreciated!

• Well, we could have $a=b \neq 0$, while the vectors are not equal, so the inequality cannot hold in general for finite $C$. – Macavity Apr 1 '15 at 1:34
• Good point, I mucked up the phrasing. Edited to exclude the trivial case where it fails. – Richard Zhou Apr 1 '15 at 1:44
• Unfortunately that issue remains pertinent. Will illustrate below: – Macavity Apr 1 '15 at 2:08

Let v = -w, and $\|\mathbf{v}\|=\|\mathbf{w}\|= R$. Then we need $$C \ge \frac{a+b}{|a-b|}R$$
The RHS now cannot be bound from above by any finite number which does not depend on $b/a$.