Comparison of equivalent norms

Question

Let $\|\cdot\|_a, \|\cdot\|_b$ be equivalent norms. Does it hold that $$\|u\|_a < \|v\|_a \implies \|u\|_b < \|v\|_b$$ My intuition says yes but I fail to prove it.

Answer 1

No. $u=(0,2)$ and $v=(1,2)$ obey $\|u\|_2 < \|v\|_2$ (Euclidean norm) while $\|u\|_\infty= \|v\|_\infty$ (max-norm), while all norms on $\Bbb R^2$ are equivalent.

Answer 2

Consider $\|.\|_1$ and $\|.\|_2$ on $\mathbb R^{2}$. Let $v=(1,0)$ and $u=(a,a)$ with $a>0$ Then $\|u\|_2<\|v\|_2$ iff $2a^{2}<1$ or $ a <\frac 1 {\sqrt2 }$. On the other hand $\|u\|_1<\|v\|_1$ iff $2a<1$ or $a<\frac 1 2$. So take $a$ between $\frac 1 2 $ and $\frac 1 {\sqrt 2}$ to get a counterexample.