We will first attempt to prove that

First let us prove that if .

Since we can find partitions of respectively such that

and

Adding the two together gives

Now there must exist parition of the form . Where the result follows since

The proof for follows by assuming the opposite and using the above.

Now to prove the integral one again consider partitions of respectively. Now once again there must exist a partition of of the form , where it follows

Taking the supremum over all partitions of the appropriate intervals gives

But it must also be true that

Taking the infimum over all the partitions of the appropriate intervals gives

Combining gives

Any criticism would be greatly appreciated!