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 $\displaystyle (X_n)_{n \geq 1}$ uniformly distributed over $\displaystyle [0,1]$

For any continuous function $\displaystyle f ~:~ [0,1]\to \mathbb{R}$, and hence bounded (since continuous over a compact set), we have :

$\displaystyle \mathbb{E}f \left(\frac{X_1+\dots+X_n}{n}\right)=\int_{\mathbb {R}^n} f \left(\frac{x_1+\dots+x_n}{n}\right) \bold{1}_{\{x_1 \in [0,1]\}}\dots \bold{1}_{\{x_n \in [0,1]\}} ~dx_1\dots dx_n$

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 :

$\displaystyle \mathbb{E}f \left(\frac{X_1+\dots+X_n}{n}\right)=\int_0^1 \dots \int_0^1 f \left(\frac{x_1+\dots+x_n}{n} \right) ~dx_1\dots dx_n$

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

$\displaystyle \frac{X_1+\dots+X_n}{n} \underset{n \to \infty}{\longrightarrow} \mathbb{E}(X_1)$ in probability, and thus in distribution.

But $\displaystyle \mathbb{E}(X_1)=\int_0^1 x ~dx=\frac 12$

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

$\displaystyle \mathbb{E}f \left(\frac{X_1+\dots+X_n}{n}\right) \underset{n \to \infty}{\longrightarrow} \mathbb{E}f \left(\frac 12\right)=f \left(\frac 12\right)$

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 :

$\displaystyle \lim_{n \to \infty} \sum_{k=0}^n \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k} f \left(\frac kn\right)=f(p)$

where $\displaystyle p \in [0,1]$ and f is a continuous function $\displaystyle [0,1] \to \mathbb{R}$

$\displaystyle \lim_{n \to \infty} \sum_{k \geq 0} e^{-\lambda n} \frac{(\lambda n)^k}{k!} f\left(\frac kn\right)=f(\lambda)$

where $\displaystyle \lambda \in (0,\infty)$ and f is a real valued continuous and bounded function defined over $\displaystyle \mathbb{R}^+$