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Thread: Partitions

  1. #1
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    Exclamation Partitions

    Let $\displaystyle [a,b] \rightarrow{}\mathbb{R}$ a bounded fuction.

    Prove that if $\displaystyle P_1$ and $\displaystyle P_2$ are partitions of $\displaystyle [a,b]$ then:

    $\displaystyle L(P_1,f) \leq{U(P_2,f)}$

    Thanks a lot
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  2. #2
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    Quote Originally Posted by osodud View Post
    Let $\displaystyle [a,b] \rightarrow{}\mathbb{R}$ a bounded fuction.
    Prove that if $\displaystyle P_1$ and $\displaystyle P_2$ are partitions of $\displaystyle [a,b]$ then: $\displaystyle L(P_1,f) \leq{U(P_2,f)}$
    To do this you need to understand a refinement of a partition.
    Prove that if $\displaystyle Q' $ is a refinement of the partition $\displaystyle Q$ then
    $\displaystyle L(Q,f) \leqslant L(Q',f) \leqslant U(Q',f) \leqslant U(Q,f) $.

    Then to do your problem find a refinement of $\displaystyle P_1\cup P_2$.
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