Ok I'm supposed to prove that if for any real x&y s.t x<y there exist a real z s.t(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Proving that a real number exists in between a real number,

Loading...

Similar Threads - Proving real number | Date |
---|---|

I A problematic limit to prove | Jan 26, 2018 |

I Proving equivalence between statements about a sequence | Feb 12, 2017 |

I Prove that ∫f(x)δ(x)dx=f(0) | Jan 22, 2017 |

I Prove ln(x) <= x-1 for positive x | Jan 15, 2017 |

Proving Cauchy Sequence Converges on Real Number Line | Mar 21, 2011 |

**Physics Forums - The Fusion of Science and Community**