However, with big-O notation the sequence can only exceed the bound in a finite prefix of the sequence, whereas the limit superior of a sequence like e−"n" may actually be less than all elements of the sequence. The only promise made is that some tail of the sequence can be bounded above by the limit superior plus an arbitrarily small positive constant, and bounded below by the limit inferior minus an arbitrarily small positive constant. The limit superior and limit inferior of a sequence are a special case of those of a function (see below). In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers. Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the complete totally ordered set [−∞,∞], which is a complete lattice. Consider a sequence formula_11 consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite). The relationship of limit inferior and limit superior for sequences of real numbers is as follows:  As mentioned earlier, it is convenient to extend formula_37 to [−∞,∞]. Then, ("x""n") in [−∞,∞] converges if and only if  in which case formula_39 is equal to their common value. (Note that when working just in formula_37, convergence to −∞ or ∞ would not be considered as convergence.) Since the limit inferior is at most the limit superior, the following conditions hold  If formula_43 and formula_44, then the interval ["I", "S"] need not contain any of the numbers "x""n", but every slight enlargement ["I" − ε, "S" + ε] (for arbitrarily small ε > 0) will contain "x""n" for all but finitely many indices "n". In fact, the interval ["I", "S"] is the smallest closed interval with this property. We can formalize this property like this: there exist subsequences formula_45 and formula_46 of formula_1 (where formula_48 and formula_49 are monotonous) for which we have  On the other hand, there exists a formula_51 so that for all formula_52  To recapitulate:   In general we have that  The liminf and limsup of a sequence are respectively the smallest and greatest cluster points.