In these cases, the concept of a Cauchy sequence is useful. Definition. Let be a real-valued sequence. We say that is a Cauchy sequence if, for any , there exists a natural number such that implies that . It can be shown that a real-valued sequence is Cauchy if and only if it is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, , is a complete metric space. In a general metric space, however, a Cauchy sequence need not converge. In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent. Uniform and pointwise convergence for sequences of functions In addition to sequences of numbers, one may also speak of sequences of functions on , that is, infinite, ordered families of functions , denoted , and their convergence properties. However, in the case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence, that need to be distinguished. Roughly speaking, pointwise convergence of functions to a limiting function , denoted , simply means that given any , as . In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely. Uniform convergence requires members of the family of functions, , to fall within some error of for every value of , whenever , for some integer . For a family of functions to uniformly converge, sometimes denoted , such a value of must exist for any given, no matter how small. Intuitively, we can visualize this situation by imagining that, for a large enough , the functions are all confined within a 'tube' of width about (that is, between and ) for every value in their domain . The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence. For example, a sequence of continuous functions (see below) is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise. Karl Weierstrass is generally credited