we can find so that if , then

Lastly, by our assumptions (assuming are finite) there exists an such that if , then .

Then we can calculate the following upper bound:

If we divide both sides of the above inequality by

and take the limit we get:

Since is arbitrary we get the upper bound:

And combining this with the lower bound gives the result.

Note that the above proof obviously fails when or (or both). To deal with these cases,