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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
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