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Math Help - infimum proof question

  1. #1
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    infimum proof question

    prove that:

    √3 = inf{ x in rationals: x > 0 and x^2 > 3}

    thank you for your time!
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  2. #2
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    Let A=\inf\{x\in \mathbb{Q}:0<x\land x^2>3\} and suposse A\neq \sqrt{3}. Then A>3 or A<3.

    Now you can use the fact that \mathbb{Q}\subset \mathbb{R}\subset \overline{\mathbb{Q}}.
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  3. #3
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    Quote Originally Posted by Abu-Khalil View Post
    Let A=\inf\{x\in \mathbb{Q}:0<x\land x^2>3\} and suposse A\neq \sqrt{3}. Then A>3 or A<3.

    Now you can use the fact that \mathbb{Q}\subset \mathbb{R}\subset \overline{\mathbb{Q}}.

    What do you mean by \overline{\mathbb{Q}}? Because if this is, as usually denoted, the algebraic closure of Q then definitely \mathbb{R} \nsubseteq\overline{\mathbb{Q}}. For example, \pi\notin\overline{\mathbb{Q}}

    Tonio
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  4. #4
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    Quote Originally Posted by tonio View Post
    What do you mean by \overline{\mathbb{Q}}? Because if this is, as usually denoted, the algebraic closure of Q then definitely \mathbb{R} \nsubseteq\overline{\mathbb{Q}}. For example, \pi\notin\overline{\mathbb{Q}}

    Tonio
    No, i mean closure in terms of limit points. You usually read as \mathbb{Q} is dense in \mathbb{R}.
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  5. #5
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    Quote Originally Posted by Abu-Khalil View Post
    No, i mean closure in terms of limit points. You usually read as \mathbb{Q} is dense in \mathbb{R}.

    Oh, I see...but then in fact \mathbb{R}=\overline{\mathbb{Q}}

    Tonio
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