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Math Help - infimum question

  1. #1
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    infimum question

    let A be a nonempty subset of positive real numbers with
    <br />
 \inf \ A \neq\ 0 .<br />
\text{ prove that } \sup \{ \frac{1}{a} \   ; a \ \in \  A  \} \ = \frac {1}{\inf A}
    Last edited by Plato; October 28th 2009 at 12:13 PM.
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  2. #2
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    Quote Originally Posted by flower3 View Post
    let A be a nonempty subset of positive real numbers with
    <br />
 \inf \ A \neq\ 0 .<br />
\text{ prove that } \sup \{ \frac{1}{a} \   ; a \ \in \  A  \} \ = \frac {1}{\inf A}
    Suppose that 0<\lambda =\inf(A). If a\in A then \frac{1}{a}\le \frac{1}{\lambda}.
    Therefore the set \left\{\frac{1}{a}:a\in A\right\} has a \sup \gamma\le \frac{1}{\lambda}.

    Suppose that \gamma <\frac{1}{\lambda}. Show that leads to a contradiction.
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