Proving Riemann Integral: Non-Negative f(x)=0 $\rightarrow \int^{b}_{a}f=0$

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SUMMARY

The discussion focuses on proving that if a bounded, non-negative function f(x) is identically zero on the interval [a, b], then the Riemann integral of f over that interval equals zero, expressed as ∫ab f(x) dx = 0. The approach involves demonstrating that both the lower sums and upper sums converge to zero as the norm of the partition approaches zero. The participants clarify that since f(x) is specified as bounded and non-negative, it implies that f(x) must indeed be the zero function across the interval.

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Homework Statement




Suppose f(x):[a,b]\rightarrow\Re is bounded, non-negative and f(x)=0. Prove that \int^{b}_{a}f=0.


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The Attempt at a Solution


I am trying to use the idea that lower sums are zero, and show that the upper sums go to zero as the norm of the partition goes to zero. That is Upper sums \leq c||P|| such that \int^{\overline{b}}_{a}f=0.

how can I prove that statement by using the idea above with the choice of c?
 
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Are you saying that f is the zero function? I.e identically zero? (I only ask because you specify that f is bounded and non-negative, which would seem to be redundant.)

Pick any partition. What are the Riemann sums for this partition? (This is trivial to work out, assuming I understand correctly that f(x)=0 for all x in [a,b]).
 

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