Under <complete convergence of random variables>, for each integer $k\ge1$ the first <Borel-Cantelli lemma> gives $\Pr(|X_n-X|>1/k\text{ infinitely often})=0$. Intersecting these <probability>-one conclusions over the countably many $k$ shows that every tolerance is eventually satisfied. Therefore \b[<complete convergence> implies <almost sure convergence>], without <independence>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 25J / c / Solution

 Solution (source code)