Can anyone help me please, I dont really have an idea.

Suppose f is bounded on a closed interval and there exists a partition P for which U(f;P) = L(f;P). Is f Riemman integrable on the closed inteval?

Thank you

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- July 25th 2008, 03:08 PM #1

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- Jul 2008
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- July 25th 2008, 08:53 PM #2
for any partition P of [a,b],

L(f;P) =< U(f;P), L(f) =< U(f)

and

L(f;P) =< L(f), U(f) =< U(f;P)

this means that

U(f) - L(f) =< U(f;P) - L(f;P)..

if epsilon is any positive real such that there exists a partition P for which U(f;P) = L(f;P) (assumption) implying U(f;P) - L(f;P) < epsilon,

then 0 =< U(f) - L(f) < epsilon, which implies U(f) - L(f) = 0.

hence U(f) = L(f), therefore f is Riemann integrable on [a,b].QED