I'm confused by the different definitions of equivalent norms. The standard one I know is that two norms (on a vector space) are equivalent if they induce equivalent metrics; two metrics are equivalent if a sequence is Cauchy w.r.t. one metric iff it is Cauchy w.r.t. the other.
It can be proven that an equivalent definition of equivalence of norms is the following: we can find real constants $A$ and $B$ such that $A|x|_1 \leq |x|_2 \leq B|x|_1$ for all $x$ in our vector space. For a reference, see e.g. Theorem 2.1 in this document.
My problem is as follows. I want to prove that a norm on $\mathbb Q$ given by $||x|| = |x|^\alpha$, where $|\cdot|$ is the Euclidean norm and $0 < \alpha \leq 1$, is equivalent to the Euclidean norm. I can prove that the above definition of $||\cdot||$ induces a metric equivalent to the Euclidean metric. I can also prove that in this case, it is impossible to find constants $A$ and $B$ such that $A||x|| \leq |x| \leq B||x||$ for all $x \in \mathbb Q$. What's going on?