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

By convergence in probability, choose recursively $n_k>n_{k-1}$ so thatThe corresponding events have a summable sequence of probabilities. The First Borel-Cantelli lemma, which requires no independence, therefore makes the probability of infinitely many of them zero. On the complementary event, $|X_{n_k}-X|\le 2^{-k}$ for all sufficiently large $k$, provingThe recursively chosen indices are strictly increasing and hence tend to infinity.