**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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