Okay, actually, this is something where my TA said "and with this example, we can see that probability surpasses analysis"

Let's consider a sequence of independent random variables uniformly distributed over

For any continuous function , and hence bounded (since continuous over a compact set), we have :

where **1** is the indicator function.

This formula is what we call "the transfer formula" (it may be derived from the Radon-Nikodym theorem - okay, it's an analysis theorem xD)

This gives :

And we know from the weak of large numbers (which can be applied here) that :

in probability, and thus in distribution.

But

Convergence in distribution implies that for any bounded & continuous function f, we have :

And this completes the proof.

Sidenotes :

- it's long because I tried to explain it as much as possible

- if you don't like probability, you surely wouldn't like this proof

- it is possible to prove, similarly, that :

where

and f is a continuous function

where

and f is a real valued continuous and bounded function defined over