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GVAR717
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This is my first time posting & I am not familiar with how to get all the correct math symbols or how to use Latex, so I did the best I could.
Suppose f is bounded on [a,b] and there is a partition P* of [a,b] for which S(f,P*)=S(f,P*). Is f Riemann integrable on [a,b]?
S(f)=sup{S(f,P*): P* is partition of [a,b]}
S(f)=inf{S(f,P*); P* is partition of [a,b]}
I know, by a theorem, that S(f)>S(f). I am trying to figure out how to show S(f)<S(f) so that I can say S(f)=S(f). I thought about choosing another partition Pe such that S(f,Pe)-S(f,Pe) would equal some epsilon value, but I don't know what value I should use or where to go next.
If this is the wrong process for this proof, I would love a hint on where to start.
Homework Statement
Suppose f is bounded on [a,b] and there is a partition P* of [a,b] for which S(f,P*)=S(f,P*). Is f Riemann integrable on [a,b]?
Homework Equations
S(f)=sup{S(f,P*): P* is partition of [a,b]}
S(f)=inf{S(f,P*); P* is partition of [a,b]}
The Attempt at a Solution
I know, by a theorem, that S(f)>S(f). I am trying to figure out how to show S(f)<S(f) so that I can say S(f)=S(f). I thought about choosing another partition Pe such that S(f,Pe)-S(f,Pe) would equal some epsilon value, but I don't know what value I should use or where to go next.
If this is the wrong process for this proof, I would love a hint on where to start.
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