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Math Help - Proof that Upper Riemann Integral > Lower Riemann Integral...?

  1. #1
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    Talking Proof that Upper Riemann Integral > Lower Riemann Integral...?

    heyyy guys....
    im new at this level of maths so i would appreciate a lil bit of help here and there hehe

    i want to show that
    lower riemann integral  \int^b_a f(x)dx \leq upper riemann integral \int^b_a f(x)dx

    i have worked this out so far but im not sure if its correct: please help me on my solution.

    proof:
    if P,Q are partitions of P[a,b] (i.e P,Q  \epsilon P[a,b] then
    L(f,P)  \leq U(f,Q)
    If we let R:=P U Q, be a common refinement of P & Q then we see that:

    L(f,P)  \leq L(f,R) &
    U(f,R)  \leq U(f,Q)

    We know L(f,R)  \leq U(f,R) so now,
    L(f,P)  \leq L(f,R)  \leq U(f,R)  \leq U(f,Q)

    This deduces that L(f,P)  \leq U(f,Q).
    Therefore,
    sup L(f,P)  \leq inf U(f,P).... which is the same as

    lower riemann integral < upper riemann integral..


    Is this proof totally correct? or is there another solution to proving this? thanks a lot xxx
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  2. #2
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    Quote Originally Posted by joanne_q View Post
    heyyy guys....
    im new at this level of maths so i would appreciate a lil bit of help here and there hehe

    i want to show that
    lower riemann integral  \int^b_a f(x)dx \leq upper riemann integral \int^b_a f(x)dx

    i have worked this out so far but im not sure if its correct: please help me on my solution.

    proof:
    if P,Q are partitions of P[a,b] (i.e P,Q  \epsilon P[a,b] then
    L(f,P)  \leq U(f,Q)
    If we let R:=P U Q, be a common refinement of P & Q then we see that:

    L(f,P)  \leq L(f,R) &
    U(f,R)  \leq U(f,Q)

    We know L(f,R)  \leq U(f,R) so now,
    L(f,P)  \leq L(f,R)  \leq U(f,R)  \leq U(f,Q)

    This deduces that L(f,P)  \leq U(f,Q).
    Therefore,
    sup L(f,P)  \leq inf U(f,P).... which is the same as

    lower riemann integral < upper riemann integral..


    Is this proof totally correct? or is there another solution to proving this? thanks a lot xxx
    Remove the redundance, and don't claim things that you are going to
    show before you show that they are true and correct the slight errors:

    proof:

    Let P,Q be partitions of [a,b] (i.e P,Q  \epsilon P[a,b]), and R be a common refinement of P & Q.

    We know L(f,R)  \leq U(f,R) hence L(f,P)  \leq L(f,R)  \leq U(f,R)  \leq U(f,Q)

    Which tells us that L(f,P)  \leq U(f,Q).

    Therefore, sup L(f,P)  \leq inf U(f,P).... which is the same as

    lower riemann integral \le upper riemann integral..

    RonL
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  3. #3
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    i would have said the same
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  4. #4
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    thanks for the reply captain.

    Quote Originally Posted by CaptainBlack View Post
    Remove the redundance, and don't claim things that you are going to
    show before you show that they are true and correct the slight errors:
    > which parts were u referring to that should be made redundant? and where were the errors please?



    Quote Originally Posted by CaptainBlack View Post
    proof:
    Which tells us that L(f,P)  \leq U(f,Q).

    Therefore, sup L(f,P)  \leq inf U(f,P).... which is the same as

    lower riemann integral \le upper riemann integral..

    RonL

    > here you showed that : (a) L(f,P)  \leq U(f,Q).
    but how does this deduce to the fact that, (b) sup L(f,P)  \leq inf U(f,P)?

    this is because the darboux sums of (b) involve only the partition P... whereas (a) involves both P and Q? ...
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  5. #5
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    Quote Originally Posted by joanne_q View Post
    thanks for the reply captain.



    > which parts were u referring to that should be made redundant? and where were the errors please?


    The proposed proof that I posted was what you had with the redundant stuff
    deleted (as the quoting what you were going to demonstrate before you had done).

    The only substantive error that I corrected was to replace < by <= in the
    last line.

    RonL
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by joanne_q View Post
    tha

    > here you showed that : (a) L(f,P)  \leq U(f,Q).
    but how does this deduce to the fact that, (b) sup L(f,P)  \leq inf U(f,P)?
    If L(f,P)  \leq U(f,Q) for all partitions P and Q of [a,b], then
    in particular:

    L(f,P)  \leq U(f,P)

    hence:

    sup L(f,P)  \leq inf U(f,P),

    where the supremum and infimum are over all partitions P.

    RonL
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  7. #7
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    thank you captainblack, you were very helpful x
    i will give you a thanks
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