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Math Help - Nested Sequences

  1. #1
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    Nested Sequences

    Let (In) be a nested sequence of closed bounded intervals. For each natural number n, let xn be in In. Use the Bolzano-Weierstrass Theorem to give a proof of the Nested Intervals property.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by frenchguy87 View Post
    Let (In) be a nested sequence of closed bounded intervals. For each natural number n, let xn be in In. Use the Bolzano-Weierstrass Theorem to give a proof of the Nested Intervals property.
    Intervals is \mathbb{R}? Do you have to use BW? It's easier to use the fact that since each interval is bounded and the diameter of the sets approaches zero (which I assume is what you want) then we get by the completeness of [tex]\mathbb{R}[/mat] that the intersesection contains a single point. Otherwise, argue that if we have [a_n,b_n] that each must converge to a point by BW. And use the diameter approaching zero to show that it the two points must be equal.
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  3. #3
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    Boundedness Removed

    I do have to use BW...
    Last edited by frenchguy87; January 24th 2010 at 10:24 PM.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by frenchguy87 View Post
    I do have to use BW...
    Ok. That's fine. But, what have you done as of now?
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