d Suppose S Q the set of rational numbers IfS is bounded above then S has a

D suppose s q the set of rational numbers ifs is

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(d) SupposeSQ,the set of rational numbers. IfSis bounded above,thenShas a rational least upper bound (sup)α.(e) IfSis a nonempty subset ofRwhich is bounded above, andTis anonempty subset ofS,thenTis bounded.(f) Ifαis the supremum of the setSRandk < α,thenkis not anupper bound forS.
4. Give the supremum and infimum (if either or both exist) of each of the followingsets. Also, if either exists, state whether it is a maximum or a minimum.(a)S=2 +(-1)nn:nN(b)S=n+ (-1)n2:nN(c)S={(-1)nn2:nN}(d)S=nsin 6 : n N o
5. LetSR, S6= Øbe bounded below and letβ= infS. Prove that ifisany positive number, then there is an elementsSsuch thatβs < β+.
6. LetSR, S6= Ø be bounded above and letα= supS. Prove that ifα /S,thenSmust have infinitely many elements.

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• Fall '08
• Staff
• Math, Empty set, Supremum, Order theory