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Math Help - I need some suggestion

  1. #1
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    I need some suggestion

    on how to prove this problem. I don't know where to start. Any help is very much appreciated.

    Suppose that a function f: [a,b] >> R is bounded and P = {x_0,x_1,...,x_n} is a partition of [a,b]. Then there exist 2 real numbers m and M such that

    m(b-a) <= L(P,f) <= S(P,f) <= U(P,f) <= M(b-a)

    for any c_k in [x_k-1,x_k]
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by EmmWalfer View Post
    on how to prove this problem. I don't know where to start. Any help is very much appreciated.

    Suppose that a function f: [a,b] >> R is bounded and P = {x_0,x_1,...,x_n} is a partition of [a,b]. Then there exist 2 real numbers m and M such that

    m(b-a) <= L(P,f) <= S(P,f) <= U(P,f) <= M(b-a)

    for any c_k in [x_k-1,x_k]
    What if you took \displaystyle M=\sup_{x\in[a,b]}f(x) and \displaystyle m=\inf_{x\in[a,b]}f(x) which you know exist by the assumption of boundedness. Then,


    \displaystyle \begin{aligned}U\left(P,f) &=\sum_{i=0}^{n}\sup_{x\in[x_{i-1},x_i]}f(x)\Delta x_i\\ &\leqslant \sum_{i=0}^{n}M \Delta x_i\\ &=M\sum_{i=0}^{n}\Delta x_i\\ &= M(b-a)\end{aligned}

    See if you can figure out how to prove the others.
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