Fundamental Theorem of Calculus
I was hoping someone could please look over this proof for any errors. My assignment is to write a formal proof of the fundamental theorem of calculus. I am instructed to use in my proof the Cauchy Criterion for integrals and the mean value theorem. I have been provided a hint that I am suppose to apply the MVT to subintervals of a partition of the interval I am integrating over, and that to complete the proof I must evaluate several quantitative relationshipships between Riemann integrals, Riemann sums, and Darboux sums. Here is my proof.
If is a differentiable, real-valued function, and is continuous and integrable, then:
Let be given. Since is integrable, there exists a partition such that:
where represent the upper and lower darboux sums.
Now consider each subinterval of , each of width . Since is differentiable, by the mean value theorem, there exists such that
Taking the sum of both sides and noting that since f(x) is continuous and thus we obtain:
Since the sum on the left is a Riemann sum, we now know:
We also know that
Adding inequalities (2) and (3) gives:
Equation (4) along with equation (1) implies that:
Since is arbitrary and thus given the right partition can be made so small that it might as well be zero, it follows that