Martingales in the limit and mils were first introduced by Mucci (1976) and Talagrand (1985), respectively. They proved that every $L^1$-bounded mil converges a.s. Recently, Luu (1999) has extended this result to sequential mils. In this note we consider sequences of random variables which are not necessarily integrable. By using a stopping time method we shall give a convergence result for their subsequences.