Results 1 to 2 of 2

Math Help - Real analysis proof

  1. #1
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303

    Real analysis proof

    Prove: a) Let \rho be a transitive relation on the interval [a,b]. If each x\in[a,b] has a neighborhood N_{x} such that u\rho\\v whenever u\in[a,x]\cap\\N_{x} and v\in[x,b]\cap\\N_{x}, then 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 x\in[a,b] be arbitrary. Since [a,b] is compact we know b=max([a,b])=sup([a,b]). Now, by our hypothesis, we can find a neightborhood of x such that if u\in[a,x]\cap\\N_{x} and v\in[x,b]\cap\\N_{x}, then u\rho\\v. So, there are two cases; first, N_{x} is an interior point of the interval; second, N_{x} is not an interior point, so a portion of the neighborhood may be outside the closed interval ( x could even equal b). In either case, we can find a transitive relation [tex]\rho[/mathp such that u\rho\\v implies 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 u\rho\\v\ iff u\leq\\v and (\exists\\n\in{\mathbb{N}} s.t. [u,v]\cap[a,b]=\emptyset.

    Thanks
    Last edited by Danneedshelp; November 18th 2009 at 07:37 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Danneedshelp's Avatar
    Joined
    Apr 2009
    Posts
    303
    Any ideas on the second part? I am stuck.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Real Analysis Proof
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: July 27th 2011, 08:08 PM
  2. Real analysis proof
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 10th 2011, 10:03 PM
  3. Real Analysis Proof
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 16th 2010, 08:27 PM
  4. Real analysis proof
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 13th 2009, 08:33 PM
  5. Real Analysis Proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 27th 2009, 03:43 PM

Search Tags


/mathhelpforum @mathhelpforum