Convergence, continuity and topology
12.1 Convergence of sequences in a metric space
We now turn to analysis on metric spaces. The definitions and results that we shall consider are straightforward generalizations of the corresponding definitions and results for the real line. The same is true of the proofs; in most cases, they will be completely straightforward modifications of proofs of results in Volume I. We shall however present the material in a slightly different order.
Suppose that (X, d) is a metric space, that is a sequence of elements of X, and that l ∈ X. We say that an converges to l, or tends to l, as n tends to infinity, and ...