Find positive numbers ,
such that
[Comments in italics]
This problem requires the following fact. For , if
and
then we may divide the smaller by the larger and the larger by the smaller to preserve the inequality, i.e.
Now
Now, for the upper bound, by the triangle inequality,
As the maximum of as
is 1; i.e. for
. We also used
By the reverse triangle inequality,
For ,
is positive [if
, then
and so
] so
We have already seen ; add
to both sides.
So we have and
hence
Now, for the lower bound, by the reverse triangle inequality:
For ,
is negative [if
, then
and so
] so
We have already seen ; add
to both sides.
Now by similar arguments to above:
So we have and
hence
Putting these together we get and
:
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