Limits of Real Sequences

Abstract

Any real number is made accessible through its rational approximations, for example, cutting off the decimals starting with (n + 1)th one. As n increases, these approximations come closer to the given real number, a process that lies at the heart of the subject of convergence. The study of many algorithms (such as the Babylonian algorithm for extracting the square root) needs some theoretical considerations of convergence and limits, which can be found in this chapter. 2.1 Convergent Sequences The notion of a sequence of real numbers is motivated by the various algorithms that make available a certain object by its successive approximations in a class of well-behaved objects. From this point of view, the main problem in connection with a sequence is its behavior for large values of indices. 2.1.1 Definition A sequence (a n) n≥0 is called convergent to the number (abbrevi-ated, a n →) if for each ε > 0, there is a natural number N such that for all n ≥ N , we have |a n − | < ε. In other words, a n → means that for each ε > 0, there is an index N such that for all n ≥ N , we have − ε < a n < < + ε. The real number , which appears in the previous definition, if it exists, is unique. In fact, if a n → and a n → , then for each ε > 0 there is an index N such that |a n − |