1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Real Analysis Intervals Proof

  1. Sep 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that I<J if and only if x<y for every x[tex]\in[/tex]I and y[tex]\in[/tex]J.

    2. Relevant equations
    I=[r,s] and J=[u,v]
    I<J means that s<u.

    3. The attempt at a solution
    Proof by contradiction: Assume that I<J if x>y for every x[tex]\in[/tex]I and y[tex]\in[/tex]J.
    Let I be the interval [r,s] and J be the interval [u,v].
    x[tex]\in[/tex]I means r[tex]\leq[/tex]x[tex]\leq[/tex]s
    y[tex]\in[/tex]J means u[tex]\leq[/tex]y[tex]\leq[/tex]v
    Since x>y, u[tex]\leq[/tex]y<x[tex]\leq[/tex]s
    Therefore u<s
    However the definiton of I<J is s<u
    This contradicts our statement
    Therefore the original statement is true, I<J if and only x<y for every x[tex]\in[/tex]I and y[tex]\in[/tex]J.

    I just wanted to make sure that this proof is correct and complete.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted