Show that sup{R element Q: r < a} = a for each a element R.

Thanks

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- April 15th 2007, 03:55 PMtbyou87Supremum Problem
Show that sup{R element Q: r < a} = a for each a element R.

Thanks - April 15th 2007, 04:00 PMThePerfectHacker
These are "Dedekind Cuts".

We need to show:

1)"a" is an upper bound.

2)"a" is least upper bound.

#1)It is an upper bound, for that is the definition of this non-empty.

#2)Say that "a" is not a least upper bound say b<a is the least upper bound. Then by the**dense**ness of Q we have that there exists an rational "r" such that b<r<a hence, r in the set. But then "b" cannot be an unpper bound. A contradiction. Thus "a" is the least upper bound.

Q.E.D.