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Math Help - 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 :
     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 :
     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 x\in B, x\in A (since B\subseteq A) and so x\leqslant\sup A (as \sup A is an upper bound for A). Hence \sup A is an upper bound for B and so \sup B\leqslant\sup A (since \sup B is the least of the upper bounds for B).

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

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