Moreover, if is pure, then and , so the equality holds in the above inequality.

The desired inequality follows from dividing the above inequality by "g"("t").

If "f" and "g" are of opposite monotonicity, then the above inequality works in the reverse way.

The reduction criterion consists of the above two inequalities.

All norms in the following sections will be norms for which the above inequalities hold (these norms are often called an energy norm).

Now, take the above inequality, let m approach infinity, and put it together with the other inequality.

If we divide both sides of the above inequality by and take the limit we get:

Finally dividing the above inequality by n!

Now define B as a positive constant that upper bounds the first term on the right-hand-side of the above inequality.

Taking expectations of the above inequality gives: