I'm spending hours trying to develop and prove the mean-square-limit.

Does anyone know where I could find an exhaustive and detailed proof online?

I would really appreciate!

Here is the concept:

Assume the finite time t is sliced into n identical partitions.

Define tj as the time up to the end of the j-th partition.

The mean-square-limit sais that

where X's are brownian motions and E is the expectation.

Limit is zero because (developing and resolving) E=O(1/n)