The statement that Xₙ gets close to 0 can mean several things. The typewriter sequence of sweeping intervals gets small in probability and on average, yet at every single outcome it keeps returning to 1, so it has no almost sure limit. A shrinking spike converges almost surely but not in mean. The sample mean of coin flips converges in every sense. Switch between the three sequences, raise n and compare what each mode asks for: a fixed outcome, the probability of a large gap, or the average size of the gap.