Probability and Measure

A sequence of real numbers an converges to a limit a if and only if, for each 2 > 0, the sequence an eventually lies within a ball of radius 2 centered at a. It’s okay if the first few (or few million) terms lie outside that ball— and the number of terms that do lie outside the ball may depend on how big 2 is (if 2 is small enough it will take millions of terms before the remaining sequence lies inside the ball). This can be made mathematically precise by introducing a letter (say, N2) for how many initial terms we have to throw away, so that an → a if and only if there is an N2 < ∞ so that, for each n ≥ N2, |an − a| < 2: only finitely many an can be farther than 2 from a. The same notion of convergence really works in any (complete) metric space, where we require that some measure of the distance d(an, a) from an to a tend to zero in the sense that it exceeds each number 2 > 0 for at most some finite number N2 of terms. Points an in d-dimensional Euclidean space will converge to a limit a ∈ Rd if and only if each of their coordinates converges; and, since there are only finitely many of them, if they all converge then they do so uniformly (i.e., for each 2 we can take the same N2 for all d of the coordinate sequences).