bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf Xn, is the largest meeting of tails of the sequence, and the outer limit, lim sup Xn, is the smallest joining of tails of the sequence. The following makes this precise. Let In be the meet of the nth tail of the sequence. That is, The sequence (In) is non-decreasing (i.e. In ⊆ In+1) because each In+1 is the intersection of fewer sets than In. The least upper bound on this sequence of meets of tails is So the limit infimum contains all subsets which are lower bounds for all but finitely many sets of the sequence. Similarly, let Jn be the join of the nth tail of the sequence. That is, The sequence (Jn) is non-increasing (i.e. Jn ⊇ Jn+1) because each Jn+1 is the union of fewer sets than Jn. The greatest lower bound on this sequence of joins of tails is So the limit supremum is contained in all subsets which are upper bounds for all but finitely many sets of the sequence. Examples The following are several set convergence examples. They have been broken into sections with respect to the metric used to induce the topology on set X. Using the discrete metric The Borel–Cantelli lemma is an example application of these constructs. Using either the discrete metric or the Euclidean metric Consider the set X = {0,1} and the sequence of subsets: The "odd" and "even" elements of this sequence form two subsequences, ({0}, {0}, {0}, ...) and ({1}, {1}, {1}, ...), which have limit points 0 and 1, respectively, and so the outer or superior limit is the set {0,1} of these two points. However, there are no limit points that can be taken from the (Xn) sequence as a whole, and so the interior or inferior limit is the empty set { }. That is, lim sup Xn = {0,1} lim inf Xn = { } However, for (Yn) = ({0}, {0}, {0}, ...) and (Zn) = ({1}, {1}, {1}, ...): lim sup Yn = lim inf Yn = lim Yn = {0} lim sup Zn = lim inf Zn = lim Zn = {1} Consider the set X = {50, 20, −100, −25, 0, 1} and the sequence of subsets: As in the previous two examples, lim sup Xn = {0,1} lim inf