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Math Help - Sequences

  1. #1
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    Sequences

    For n \in {\mathbb {N}}, define I_n = [{a_n, b_n}] to be a sequence of closed and bounded intervals such that I_1\supseteq {I_2} \supseteq {I_3} ...

    Prove that there is a real number x such that x \in I_n for all n i.e.

         {^\infty}
    \bigcap {I_n}{\neq}{\oslash}
    _{i=1}
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by yusukered07 View Post
    For n \in {\mathbb {N}}, define I_n = [{a_n, b_n}] to be a sequence of closed and bounded intervals such that I_1\supseteq {I_2} \supseteq {I_3} ...

    Prove that there is a real number x such that x \in I_n for all n i.e.

         {^\infty}
    \bigcap {I_n}{\neq}{\oslash}
    _{i=1}
    This is a sequence of closed subsets of [a_1,b_1] which have the FIP, apply [a_1,b_1]'s compactness.
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  3. #3
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by yusukered07 View Post
    For n \in {\mathbb {N}}, define I_n = [{a_n, b_n}] to be a sequence of closed and bounded intervals such that I_1\supseteq {I_2} \supseteq {I_3} ...

    Prove that there is a real number x such that x \in I_n for all n i.e.

         {^\infty}
    \bigcap {I_n}{\neq}{\oslash}
    _{i=1}
    Since \{a_{n}\}_{n\in\mathbb{N}} is nondecreasing and \{b_{n}\}_{n\in\mathbb{N}} is increasing,
    we have a:=\lim\nolimits_{n\to\infty}a_{n} and b:=\lim\nolimits_{n\to\infty}b_{n}.
    Then (a+b)/2\in[a,b]=\cap_{n\in\mathbb{N}}[a_{n},b_{n}].
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