For all consider the quadratic .

By Descartes rule of signs this has exactly one positive root, which I will call , which depends only on (if you are not happy with Descartes rule of signs here you can find the positive root explicitly if you prefer).

Now suppose , then:

.

But:

so:

But by construction .

Hence we have proven that for all there exists a such that implies that , so we have proven that:

RonL