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Math Help - Help with nested interval proof

  1. #1
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    Help with nested interval proof

    The proof I'm working on states:

    Let [a_{n},b_{n}] be nested intervals such that b_{n}-a_{n} is null. Let x_{0} be the real number satisfying \bigcap_{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}. Let \delta >0. Prove there is an N, such that [a_{N},b_{N}]\subseteq (x_{0}-\delta ,x_{0}+\delta ).

    Here is the direction that I think I need to go but I'm not sure...
    Since \bigcap_{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}, and x_{0}-\delta <x_{0}<x_{0}+\delta then there must be an interval [a_{N},b_{N}] that is also contained in (x_{0}-\delta ,x_{0}+\delta ). But all that might be completely wrong.
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  2. #2
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    Here some useful observations.
    \left( {\forall n} \right)\left[ {a_n  \leqslant x_0  \leqslant b_n } \right].

     \left( {a_n } \right) \to x_0 \;\& \;\left( {b_n } \right) \to x_0

    If \delta>0 find an \left( N \right)\left[ {x_0  - a_N  < \frac{\delta }{2}\;\& \,b_N  - x_0  < \frac{\delta }{2}} \right]
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    The proof I'm working on states:

    Let [a_{n},b_{n}] be nested intervals such that b_{n}-a_{n} is null. Let x_{0} be the real number satisfying \bigcap_{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}. Let \delta >0. Prove there is an N, such that [a_{N},b_{N}]\subseteq (x_{0}-\delta ,x_{0}+\delta ).

    Here is the direction that I think I need to go but I'm not sure...
    Since \bigcap_{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}, and x_{0}-\delta <x_{0}<x_{0}+\delta then there must be an interval [a_{N},b_{N}] that is also contained in (x_{0}-\delta ,x_{0}+\delta ). But all that might be completely wrong.
    Suppose not, then [a_N,b_N]\supseteq (x_{0}-\delta,x_0+\delta) for all N\in\mathbb{N} so that \displaystyle (x_0-\delta,x_0+\delta)\subseteq\bigcap_{n\in\mathbb{N}  }[a_n,b_n]...but what's wrong with that?
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  4. #4
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    Well, we stated that \bigcap_{n\in\mathbb{N}}[a_n,b_n] contained only one element, x_{0}, but the way you stated it in the last line of your last post, \bigcap_{n\in\mathbb{N}}[a_n,b_n] contains numerous point is the interval (x_0-\delta,x_0+\delta).
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    Well, we stated that \bigcap_{n\in\mathbb{N}}[a_n,b_n] contained only one element, x_{0}, but the way you stated it in the last line of your last post, \bigcap_{n\in\mathbb{N}}[a_n,b_n] contains numerous point is the interval (x_0-\delta,x_0+\delta).
    That's right.
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