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). The case of sequences of real numbers 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. Interpretation Consider a sequence consisting of real numbers. Assume that the limit superior and limit inferior are real numbers (so, not infinite). The limit superior of is the smallest real number such that, for any positive real number , there exists a natural number such that for all . In other words, any number larger than the limit superior is an eventual upper bound for the sequence. Only a finite number of elements of the sequence are greater than . The limit inferior of is the largest real number such that, for any positive real number , there exists a natural number such that for all . In other words, any number below the limit inferior is an eventual lower bound for the sequence. Only a finite number of elements of the sequence are less than . Properties The relationship of limit inferior and limit superior for sequences of real numbers is as follows: As mentioned earlier, it is convenient to extend to Then, in converges if and only if in which case is equal to their common value. (Note that when working just in convergence to or would not be considered as convergence.) Since the limit inferior is at most the limit superior, the following conditions hold If and , then the interval need not contain any of the numbers but every slight enlargement for arbitrarily small will contain for all but finitely many indices In fact, the interval is the smallest closed interval with this property. We