Let be a reduced form such that . We have the following inequalities:

Set Then , and and since we have and hence the RHS is an increasing function ofR, so the inequality must be true at least for , thus giving In particular, this impliesWe also obtain from above

and thus giving the inequalities and , or in other words and . The same inequalities can be obtained directly from (5).

Hence we obtain:

Now an easy computation shows that the maximum of subject to

and is equal to , thus giving all but the last inequality of the lemma.x,ynonnegative and is attained forFor the last inequality using and , we obtain

thus proving all the inequalities of the lemma.

Note that in the first four inequalities, we have used only the bound for

A+Band not the stronger bound for . However it can be checked that this would not improve the final bounds. However, this is not the case for the next two inequalities.