expression , which denotes the distance between and . Applications and important results If and are convergent sequences, then the following limits exist, and can be computed as follows: for all real numbers , provided that for all and Moreover: If for all greater than some , then . (Squeeze Theorem)If is a sequence such that for all then is convergent, and . If a sequence is bounded and monotonic then it is convergent. A sequence is convergent if and only if all of its subsequences are convergent. Cauchy sequences A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is Cauchy characterization of convergence for sequences: A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy. In contrast, there are Cauchy sequences of rational numbers that are not convergent in the rationals, e.g. the sequence defined by x1 = 1 and xn+1 = is Cauchy, but has no rational limit, cf. here. More generally, any sequence of rational numbers that converges to an irrational number is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers. Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called complete metric spaces and are particularly nice for analysis. Infinite limits In calculus, it is common to define notation for sequences which do not converge in the sense discussed above, but which instead become and remain arbitrarily large, or become and remain arbitrarily negative. If becomes arbitrarily large as , we write In this case we say that the sequence diverges, or that it converges to infinity. An example of such a sequence is . If becomes arbitrarily negative (i.e. negative and large in magnitude) as , we write and say that the sequence diverges or converges to negative infinity. Series A series is, informally speaking, the sum of the terms of a sequence. That is, it is an expression of the form or , where is a sequence of real or complex numbers. The partial sums of a series are the expressions resulting from replacing the infinity symbol with a finite number, i.e. the Nth partial sum of the series is