In other words the strategy is that we need to consider a set....show that is the supremum of that set...and then show uniqueness?
Fix and . Prove that there is a unique real such that .
So to show uniqueness we need to consider the following: if and then . But first we want to show that is monotonically increasing.
So for any positive integer , . We can show this by factoring the LHS as . So . Or . From here what do we do?
Let us consider the slightly different problem of showing that there is a unique . So we fix . Then uniqueness follows because . And to show existence we consider the set of 's such that .
So we use this same set in our problem. But we are trying to use this strategy: .