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Thread: supremum and infimum

  1. #1
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    supremum and infimum

    Let A be a bounded subset of R "REAL NUMBERS" B a nonempty subset of A .show that :
    $\displaystyle inf A \leq inf B \leq supB \leq sup A $

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  2. #2
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    Quote Originally Posted by flower3 View Post
    Let A be a bounded subset of R "REAL NUMBERS" B a nonempty subset of A .show that :
    $\displaystyle inf A \leq inf B \leq supB \leq sup A $


    First let's prove inf(B) <= sup(b)
    For any x in B, by defintion
    inf(B)<=x<=sup(B)

    All you need to observe now is inf(A) is lower bound of B and sup(A) upper bound of B. Now invoke the definitions of inf,sup
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  3. #3
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    For all $\displaystyle x\in B,$ $\displaystyle x\in A$ (since $\displaystyle B\subseteq A)$ and so $\displaystyle x\leqslant\sup A$ (as $\displaystyle \sup A$ is an upper bound for $\displaystyle A).$ Hence $\displaystyle \sup A$ is an upper bound for $\displaystyle B$ and so $\displaystyle \sup B\leqslant\sup A$ (since $\displaystyle \sup B$ is the least of the upper bounds for $\displaystyle B).$

    A similar argument shows that $\displaystyle \inf A\leqslant\inf B.$

    Finally, as $\displaystyle B\ne\O,$ let $\displaystyle b\in B.$ Then $\displaystyle \inf B\leqslant b\leqslant\sup B$ (since $\displaystyle \inf B$ and $\displaystyle \sup B$ are a lower and an upper bound for $\displaystyle B$ respectively).
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