Be $X_1, X_2, X_3, X_4, \dots $ a sequence of uniformly limited independent random variables. Show that $S_n = X_1 + X_2 + \dots + X_n$ satisfies the Law of Large Numbers. My work, so far:
If $X_j$ goes to $U[a_j,b_j]$, so $\Omega_{X_j}(t)=\frac{e^{itb_j}-e^{ita_j}}{it}$, where $\Omega_A (t)$ is the characteristic function of an A random variable.
$$\Omega_{S_n}(t)=\frac{e^{it \sum^n b_j}-e^{it \sum^n a_j}}{it^n},\qquad \Omega_{S_n/n}(t)=\Omega_{S_n}(t/n)=\frac{e^{it \bar{b}}-e^{it \bar{a}}}{(\frac{it}{n})^n}$$