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Math Help - supremum and infimum

  1. #1
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    supremum and infimum

    Hello!
    can someone help me to prove this

    Let A*B={a*b;a,b>0 a is element of A,b element of B.Prove that:
    inf(AB)=inf(A)inf(B)
    sup(AB)=sup(A)sup(B)

    i've done the proof with inf(A+B)= inf (A) + inf(B) and inf(-A)= -inf(A) but i can't prove the problem above
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  2. #2
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    Quote Originally Posted by the prince View Post
    Hello!
    can someone help me to prove this

    Let A*B={a*b;a,b>0 a is element of A,b element of B.Prove that:
    inf(AB)=inf(A)inf(B)
    sup(AB)=sup(A)sup(B)

    i've done the proof with inf(A+B)= inf (A) + inf(B) and inf(-A)= -inf(A) but i can't prove the problem above
    We have  ab\leq \sup AB for all  a\in A, b\in B
    Fix  a. Now  b\leq \frac{1}{a} \cdot \sup AB for all  b\in B.
    But then  \sup B\leq \frac{1}{a} \cdot \sup AB.
    i.e.  a\leq \frac{1}{\sup B} \cdot \sup AB.
    Now this is true for all  a\in A.
    Hence  \sup A\leq \frac{1}{\sup B} \cdot \sup AB

    .
    i.e.  \sup A\cdot \sup B\le \sup AB

    We also have : for all a,b,belonging to A ,B ab\leq SupASupB

    HENCE Sup(AB) \leq SupASupB

    Thus Sup(AB)=SupASupB
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