Definitions of equivalent norms 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?
 A: The problem comes from the fact that the results you are quoting only hold if you consider a norm over a vector space, as defined at en.wikipedia.org/wiki/Norm_(mathematics)#Definition (not a norm in a field, as you quoted in the answer I deleted since I had misunderstood your question: encyclopediaofmath.org/index.php/Norm_on_a_field).
Indeed, if you want to prove that:
(i) Two norms $N_{1}$ and $N_{2}$ (on a vector space) are equivalent according to the definitions in your first paragraph, i.e. that for every sequence $(x_{n})$, $\lim_{m,n \rightarrow +\infty} N_{1}(x_{m}-x_{n}) = 0 \Leftrightarrow \lim_{m,n \rightarrow +\infty} N_{2}(x_{m}-x_{n}) = 0$
$\Leftrightarrow$
(ii) $\exists A,B \in \mathbb{R}, \forall x, A.N_{1}(x) \leq N_{2}(x) \leq B.N_{1}(x)$
Then you need the absolute homogeneity of the norm (which does not hold for a norm in a field where you have some kind of sub-multiplicity instead).
Proof of the usual case (norm of a vector space) is rather easy by contradiction (assuming (ii) does not hold, you can build a sequence that is Cauchy for one norm but not the other). This proof also shows why you need that absolute homogeneity.
