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Math Help - upper / lower integral proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    upper / lower integral proof

    I am trying to prove the following lemma:

    For any bounded function f on [a,b], it is always the case that U(f)\geq\\L(f), where U(f)=inf\{U(f,P):P\in{Q}\} and L(f)=sup\{L(f,P):P\in{Q}\}. For clarification, Q is the collection of all possible partitions of the interval [a,b].

    This seems trivial, but I am not sure how to construct a solid proof. There are a few lemmas I am temped to use, but when I try to use them, I basically just end up restating the problem statement.

    Any guidance would be great,

    thanks
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  2. #2
    Member Black's Avatar
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    It follows from the fact that if P,R \in Q, then L(f,P) \le U(f,R).
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  3. #3
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by Black View Post
    It follows from the fact that if P,R \in Q, then L(f,P) \le U(f,R).
    That fact alone is proof enough...?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Danneedshelp View Post
    That fact alone is proof enough...?
    Clearly. What aspect of it are you finding difficult?
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  5. #5
    Member Black's Avatar
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    If L(f,P) \le U(f,R), \, \forall P,R \in Q, then surely \text{sup}_{P \in Q}L(f,P) \le U(f,R) \Longrightarrow \text{sup}_{P \in Q}L(f,P) \le \text{inf}_{P \in Q} \,U(f,P).
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