Deducing equivalence between norms from simple condition Let $||\cdot||,\;||\cdot||'$ be norms on $V$. Suppose for some $a,b>0$ we have:
$$||x||<a\Rightarrow ||x||'<1\Rightarrow ||x||<b$$
Show that $||\cdot||,\;||\cdot||'$ are Lipschitz equivalent.

I don't think this should be hard, but I am completely stuck. My idea was to first show that the norms are equivalent on the ball $\{x:\;||x||<a\}$ but I am just turning in circles with manipulations. A gentle nudge would be really appreciated.

Hint.
For $x \in V$, what is a collinear vector $y$ to $x$ such that $\Vert y \Vert <a$? What happens then to $\Vert y \Vert^\prime$?
And vice versa?
