Define a polynomial

such that and have opposite signs. Prove there is some such that .

* Proof. * Since

we see that has the same sign as .

Next,

Then we claim that for sufficiently large ,

hence, will have the same sign as for sufficiently large , since and so

(So, when we multiply a positive number by the result will have the same sign as .)

So, we need to show that the claimed term is indeed positive. First,

This is true since for each term in the sum

Now, since we are showing there is sufficiently large such that our claim is true, we let

Since , we know for all , so

This proves our claim, and so has the same sign as for sufficiently large . Hence, and have different signs (since and had different signs by assumption); thus, there is some such that