Fundamental Theorem of Calculus

Hi everyone,

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:

**(1)**

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

or equivalently,

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:

**(2)**

We also know that

Equivalently,

**(3)**

Adding inequalities (2) and (3) gives:

**(4)**

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