You mention two sequences a_n and b_n where a_n/b_n = L > 0 for all n. So a_n = L*b_n where L is fixed. Now you want to show if a_n converges then so does b_n and if it diverges then so does the other.
The easiest way IMO is to just put sigma signs and sum the terms to infinity and then take the L out of the series and then you have an expression where A = L*B where A is the series for a_n, B is the series for b_N and L is a constant > 0.
The rigorous way of answering this involves the delta-epsilon stuff but there are some basic identities that show if X is a convergent series (in sigma form) then cX is also convergent as well. The same holds for cX being divergent if X is divergent (but c must be a constant that is non-zero).
Again I don't know what expectations you are under but you show the above two (i.e. stuff related to cX where X is convergent/divergent and c is a non-zero constant) then that's your proof.