- #1

torquerotates

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x<z<y

I'm supposed to use the following;

Axiom: every nonempty subset S of real numbers which is bounded above has a supremum: that is there is a real number B s.t B=sup(S)

1) every nonempty subset S that is bounded below has a greatest lower bound; that is there is a real number L s.t L=inf(S)

2) The set P of positive integers (i.e 1,2,3,4..n) is unbounded from above.

3) For every real number x there exists a positive integer n s.t n>x

4) If x>0 and y is an arbitrary real number, there exists a positive integer n such that nx>y

5) If three real numbers a,x,y satisfy; a=< x=<a + y/n for any n>=0, then x=a

6) If x has a supremum, then for some x in S we have x>sup(S)-h

7) If x has an infimum, then for some x in S we have x<inf(S)+h

8) Given 2 nonempty subsets S and T of R such that s=<t Then for every s in S and t in T, S has supremum and T has an infimum, and they satisfy sup(S)=<inf(T)

really I have no clue how to start this problem. Alot of the inequalities are useless because they only involve integer n. I'm looking for a real z.