Inductive proof on increasing coefficients

Question

If I have two polynomials, $$a_nx^n+....+a_1x+a_0$$ and $$ b_nx^n+....+b_1x+b_0$$ and I want to show that the rate of which the coefficients are increasing, start from the $x^n$ term in the first equation is always greater than the second. For the case I am looking at I can show the base case always holds. But I am stuck on the inductive step. So I suppose for some k that $$\dfrac{a_{k-1}-a_k}{a_{k-2}-a_{k-1}}>\dfrac{b_{k-1}-b_k}{b_{k-2}-b_{k-1}}$$ I'm unable to show that it holds for the next one