# Math Help - Need help

1. ## Need help

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

2. Originally Posted by Green03
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
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