I will assume that you have done the base case and that we assume this is true for some

So we need to show that:

,

well:

and by supposition , so:

So if we can prove that we are done, but is equvalent to . So to compete the proof of the induction step you need to show that:

has no real root greater than or equal to (this is sufficient as for greater than its largest real root the cubic is positive).

This you do by making the substitution , then using Descartes rule of signs to show that this has no positive roots.

(In fact the only real root of is near )

RonL