The last two weeks, we have been dealing with the Fundamental theorem of calculus and it's proofs. Both me and Tom created proofs that hang on annoying technicalities and because of that they do not hold.
One of the problems would be solved if we could prove an innocent statement about nested limits.
I have proven it in the following document, but this proof only holds if f is continuous, which is not the case in our proofs, I will look some more.
The proof does seems trivial, but we are realizing more and more the importance of the tiniest details in the relationships between continuousness, differentiability, integrability, and it is not always clear, specially in elementary calculus books where proofs are given a flimsy and vague treatment, we seem to be heading straight into analysis whether we like it or not (and we do!!)
Here is the same tiny proof as a pdf: http://jadnohra.net/release/math/nested_limits.pdf