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Math Help - Nested Intervals and IVT

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    Nested Intervals and IVT

    I could really use some help. Not really sure how to get started.

    Suppose f(a,b)< 0 and f(c,d)>0. Construct nested intervals [a_{n},b_{n}] and [c_{n},d_{n}] such that f(a_{n},c_{n})\leq 0 and f(b_{n},d_{n})>0. Then show f(x_{0},y_{0})=0 if \bigcap _{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\} and \bigcap _{n=1}^\infty [c_{n},d_{n}]=\{y_{0}\}.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    I could really use some help. Not really sure how to get started.

    Suppose f(a,b)< 0 and f(c,d)>0. Construct nested intervals [a_{n},b_{n}] and [c_{n},d_{n}] such that f(a_{n},c_{n})\leq 0 and f(b_{n},d_{n})>0. Then show f(x_{0},y_{0})=0 if \bigcap _{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\} and \bigcap _{n=1}^\infty [c_{n},d_{n}]=\{y_{0}\}.
    What is this f just some continuous map f:[0,1]\times[0,1]\to\mathbb{R}? What are a,b,c,d? I think you need to type up the full question.
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    f is a continuous function on some interval I where a,b,c,d\in I.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    f is a continuous function on some interval I where a,b,c,d\in I.
    Ok? Then what does f(x,y) mean? Unless you meant f\left((a,b)\right) but then the inequality signs used are meaningless.
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    There was a typo to this problem initially. It should read...
    Suppose f(a,c)<0 and f(b,d)>0.

    I'm guessing we need to use the Intermediate Value Theorem to show that if there exists f(x_0,y_0)=0 then f(a,c)<f(x_0,y_0)<f(b,d). And then somehow use this to show that \bigcap _{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\} and \bigcap _{n=1}^\infty [c_{n},d_{n}]=\{y_{0}\}??
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    There was a typo to this problem initially. It should read...
    Suppose f(a,c)<0 and f(b,d)>0.

    I'm guessing we need to use the Intermediate Value Theorem to show that if there exists f(x_0,y_0)=0 then f(a,c)<f(x_0,y_0)<f(b,d). And then somehow use this to show that \bigcap _{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\} and \bigcap _{n=1}^\infty [c_{n},d_{n}]=\{y_{0}\}??
    You have yet to answer my question. What does f(a,c)<0 mean?
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    Well, my professor never told us but judging by the position of the proof in my notes I think we are trying to help prove the following theorem:

    If I and J are intervals and f:I\times J\rightarrow \mathbb{R} is continuous, then the range of f is an interval.

    So is it possible that f(a,b) is an interval??
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    Well, my professor never told us but judging by the position of the proof in my notes I think we are trying to help prove the following theorem:

    If I and J are intervals and f:I\times J\rightarrow \mathbb{R} is continuous, then the range of f is an interval.

    So is it possible that f(a,b) is an interval??
    But isn't it more likely given what you just said that you're suppose to consider (a,b)\in I\times J?
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    Ok well if we consider (a,b)\in I\times J could you help me get started?
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  10. #10
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    Ok well if we consider (a,b)\in I\times J could you help me get started?
    I'm sorry, no. Until I understand what's going on, I can't say anything intelligible.
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