exhibit a sequence that converges in one but not the other.
Does (p/q), (p/q)^2, (p/q)^3, ... work? As clearly if this is (x_n) then I have d_p(x_n, 0) = p^{-n} -> 0, and d_q(x_n,0) = q^{n} which does not converge.
What is the reasoning now, if the above is correct, behind this being a reason for the two metrics to be non-equivalent? I apologise for the rather ignorant questioning.
Edit: I am assuming that this would mean that the identity map is discontinuous in at least one direction, which would be the reason for the non-equivalence; however I am interested here.