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Thread: Real analysis proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    Real analysis proof

    Prove: a) Let $\displaystyle \rho$ be a transitive relation on the interval $\displaystyle [a,b]$. If each $\displaystyle x\in[a,b]$ has a neighborhood $\displaystyle N_{x}$ such that $\displaystyle u\rho\\v$ whenever $\displaystyle u\in[a,x]\cap\\N_{x}$ and $\displaystyle v\in[x,b]\cap\\N_{x}$, then $\displaystyle a\rho\\b$.

    b) Use the above result to prove Cantors Nested Intervals Property (the intersection of a family of closed sets is non-empty).


    A: Here is kind of an outline to part (a):

    Let $\displaystyle x\in[a,b]$ be arbitrary. Since $\displaystyle [a,b]$ is compact we know $\displaystyle b=max([a,b])=sup([a,b])$. Now, by our hypothesis, we can find a neightborhood of $\displaystyle x$ such that if $\displaystyle u\in[a,x]\cap\\N_{x}$ and $\displaystyle v\in[x,b]\cap\\N_{x}$, then $\displaystyle u\rho\\v$. So, there are two cases; first, $\displaystyle N_{x}$ is an interior point of the interval; second, $\displaystyle N_{x}$ is not an interior point, so a portion of the neighborhood may be outside the closed interval ($\displaystyle x$ could even equal $\displaystyle b$). In either case, we can find a transitive relation [tex]\rho[/mathp such that $\displaystyle u\rho\\v$ implies $\displaystyle a\rho\\b$, since we know [tex]b=sup([a,b]).

    I am having troubles formalizing the proof and organizing my thoughts as you may be able tell.

    b) I am stuck on this one even though my teacher offered the hint, "consider the relation $\displaystyle u\rho\\v\$ iff $\displaystyle u\leq\\v$ and $\displaystyle (\exists\\n\in{\mathbb{N}}$ s.t. $\displaystyle [u,v]\cap[a,b]=\emptyset$.

    Thanks
    Last edited by Danneedshelp; Nov 18th 2009 at 07:37 AM.
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  2. #2
    Senior Member Danneedshelp's Avatar
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    Any ideas on the second part? I am stuck.
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