Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / A / 1 / b / iv / Solution

Let $U$ have the uniform distribution on $(0,1)$, set $Y=1/U$, and take $X_n=Y/n$. The variable $Y$ is finite almost surely, so $X_n\to 0$ almost surely. For any $ϵ>0$ and all sufficiently large $n$,The harmonic series diverges, soThis dependent sequence fails complete convergence. All its members are scalar multiples of the same nonconstant random variable, which explains why the independence hypothesis cannot be dropped.