I am looking for an approximate form of the Fundamental Theorem that looks like: let be on ; for every , whenever is a partition of s.t. (some term depending on ), .

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Let be a partition of , , and write for its mesh size. denotes a Riemann sum of on this partition.

Now:

The first term can be made by chusing fine enough (this is the definition of integrability), so I ignore it and look at the second term. I can make it vanish by chusing the "right" but suppose I cannot chuse these and can control only the mesh size of .

Using the formula where the error term is as ...

I want to show this last sum can be made arbitrarily small by shrinking . I do not know how to control the error terms. Can anyone help?