Re: The Law of the Excluded Middle again (long)


On Dec 1, 11:38 am, Angus Rodgers <twir...@xxxxxxxxxxx> wrote:
|A recent thread asked for a "slick" proof that x^y + y^x > 1
|for all positive real numbers x and y; and a reference was
|given to a proof of this by Wade Ramey in sci.math in 2003.

You proved this by cases, by proving it on various subsets
S1,...,Sn of S = {(x,y) : x,y>0}. It's true that the claim
that S1,...,Sn cover S is nonconstructive. However, the set
of points where it holds is an open region, and S is the
only open region in S that contains all the S1,...,Sn
(constructively). It's a dense union of closed sets (closed
relative to S). So the argument doesn't really need much of
a change to make it constructive.

Keith Ramsay
.

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