Hello all, I'm having trouble proving this theorem.

Suppose f is Darboux integrable, then for all \epsilon > 0 there exists a \delta > 0 such that mesh (P) < \delta implies U_p(f) - L_p(f) < \epsilon.


Proof:

Let \epsilon > 0 , and suppose f is Darboux Integrable.

If f is a constant we are done, if not pick  mesh (P) < \frac{\epsilon}{\sum\limits_{k=1}^n M_k - m_k}, where k is the nth sub-interval of the partition P. M_k is the supremum of the function in that respective sub-interval, m_k is the infimum.

It follows that,

\vert U(f) - L(f) \vert < \vert U_p (f) - L_p (f) \vert < \epsilon .

However, my instructor said that, I cannot pick that delta because the suprema and infima of the sub-intervals depend on the partition. I left out some steps, because the problem with the proof is the choice of delta.

Any thoughts on this? Thank you