Letbe a reduced form such that
. We have the following inequalities:
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SetThen
, and
and since
we have
and hence the RHS is an increasing function of R, so the inequality must be true at least for
, thus giving
In particular, this implies
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We 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:
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Now an easy computation shows that the maximum of
subject to x, y nonnegative and
is attained for
and is equal to
, thus giving all but the last inequality of the lemma.
For the last inequality using
and
, we obtain
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thus proving all the inequalities of the lemma.
Note that in the first four inequalities, we have used only the bound for A+B and 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.