Definition of Equivalent Norms Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$.  I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying space $X$ has the same topology with respect to either norm.  Maybe it preserves even more properties than this too.  
But I'm finding it very difficult to use this property when doing proofs or problems because although it's very simple to state, I don't immediately see what it is saying.  In comparison, when you define 'equivalence' in other settings, like in the definition of an isomorphism of abelian groups or a continuous map, it's very clear that a certain operation or object is being preserved as you pass across a map.  
To be concrete, here are my questions: 
(1) Is there another way to characterize when norms are equivalent that might provide more intuition for what it says about is being preserved 
and (2) is there a way to show that this definition is the one you want by starting with something more fundamental (like saying that the norms induce the same topology) and then proving that it's equivalent to the stated definition?  
Any intuition for the definition would be helpful for either of these questions.   

Suppose you have a sequence which converges in $G $. The lower bound implies it converges in $F $ to the same limit.
Suppose you have a sequence which does not converge in $G $. The upper bound implies it does not converge in $F $.
That's all you need, since metric spaces are sequential spaces.
