3 - Sequences in Metric Spaces

Summary

The Limit of a Sequence

The reader must have seen in an elementary course of real analysis that the study of convergence of real sequences is carried out primarily to analyse the convergence of several types of infinite series of real numbers which, in turn, occur as solutions of differential equations occurring in many practical applications. As a further application, the notion of continuity of a function is characterised in terms of sequences, viz., a function f is continuous at x if and only if every sequence 〈x_n〉 converging to x implies that 〈f(x_n)〉 converges to f(x). In metric spaces, the notion of convergence of a sequence plays an enhanced role. We have already seen several metric spaces of sequences. The notion of convergence of a sequence in a metric space has been introduced in Section 2.6. To study it in greater detail, we make it explicit in the following definition.

Definition 3.1.1 Let (X, d) be a metric space, and 〈x_n〉 be a sequence in X. Then 〈X_n〉 is said to converge to a point x∈X if for each real number ε>0, there exists an m∈ℤ₊ such that d(x_n, x) <ε for all n≥m.

The point x in the above definition is called a limit of the sequence 〈x_n〉, and we write lim x_n = x or simply as x_n → x, or as 〈x_n〉 → x.