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Math Help - partition/darboux sums question

  1. #1
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    partition/darboux sums question

    I was wondering, given a partition P and a refinement of P, namely Q, and a bounded function f:[a,b] --> R,
    how can I prove that U(Q,f)<=U(P,f)?
    Here is what I have, Im hoping someone can confirm that its right or let me know where I went wrong??...
    Proof:
    Let P\subseteq Q.
    Suppose P={ x_0,x_1,...,x_n} and let Q be a refinement of P such that Q contains one additional point of [a,b] say c where x_{i-1}\leq c \leq x_{i}.
    let r_1=sup{f(x)| x\in[x_{i-1},c]} and
    let r_2=sup{f(x)| x\in[c,x_{i}]}.
    let M_{i}=max{ r_1,r_2}. [should this be min??]
    Then U(P,f)= \sum_{k=1}^{n}M_k\Delta x_k
    = \sum_{k=1}^{i-1}M_k(x_k-x_{k-1})+M_i(x_i-x_{i-1})+\sum_{k=i+1}^{n}M_k(x_k-x_{k-1})
    \geq \sum_{k=1}^{i-1}M_k(x_k-x_{k-1}) + r_1(c-x_{i-1})+r_2(x_i-c)+\sum_{k=i+1}^{n}M_k(x_k-x_{k-1})=U(Q,f).
    Last edited by dannyboycurtis; December 5th 2009 at 07:06 PM.
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