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Math Help - supremum and infinum

  1. #1
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    supremum and infinum

    Theorem

    E \subseteq \mathbb{R}, E \not = \varnothing

    sup E exists \Leftrightarrow inf (-E) exists

    Proof:

    "\Rightarrow"

    supE exists

    Now we show that -supE=inf (-E).
    Show that
    1. -sup E is a lower bound of -E.
    2. if s is a lower bound of -E \Rightarrow s \leq -sup E.

    1.
    \because sup E is an upper bound of E
    \therefore x \leq sup E, \forall \in E \Rightarrow -x \geq -sup E, \forall x \in E

    \therefore -sup E is a lower bound of -E (line added for correction 6:30 a.m. Sep 12)

    2. Suppose that s is a lower bound of -E

    suppose not \Rightarrow s > -sup E \Rightarrow -s < sup E

    on the other hand

    -x \geq s, \forall x \in E <------(Could someone tell me how this come about.)

    \therefore x \leq -s

    Hence,
    -s is an upper bound of E \rightarrow\leftarrow

    By 1. & 2, inf (-E) \exists & inf (-E) = sup E
    Last edited by novice; September 12th 2010 at 05:32 AM.
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  2. #2
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    I donít follow what you posted.
    Suppose that s = \sup (E) then if y\in -E then -y\in E.
    This means that -y\le s or y \ge -s.
    This shows that - s \le \inf ( - E).
    Suppose that - s < \inf ( - E) then s>-\inf(-E).
    That means \left( {\exists z \in E} \right)\left[ { - \inf ( - E) < z \le s} \right] or -z<\inf(-E).
    What is wrong with that?
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  3. #3
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    Quote Originally Posted by Plato View Post
    I donít follow what you posted.
    Suppose that s = \sup (E) then if y\in -E then -y\in E.
    This means that -y\le s or y \ge -s.
    This shows that - s \le \inf ( - E). <----This part makes a whole world of a difference.
    Suppose that - s < \inf ( - E) then s>-\inf(-E).
    That means \left( {\exists z \in E} \right)\left[ { - \inf ( - E) < z \le s} \right] or -z<\inf(-E).
    What is wrong with that?
    My original post was from lecture notes of some school of mathematics I found on the web--I don't remember where I got it. There is a missing link in the proof which made it hard to understand.

    Since you said you could not follow it, you made me feel better.

    Your instructions are superb, which I am very glad that you are on board.
    I am not a math major, so I don't have access to a real math professor, but thanks to MHF for giving me access to a teacher like you.

    Thank you, sir.
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