Numerical Methods in Computational Finance. Daniel J. Duffy

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      A simple example is to show that the sequence converges to 0. To this end, let be a positive real number. Then there exists a positive integer such that whenever .

      Definition 1.5 A sequence of elements of an ordered field F is called a Cauchy sequence if for each in F there exists a positive integer such that:

      In other words, the terms in a Cauchy sequence get close to each other while the terms of a convergent sequence get close to some fixed element. A convergent sequence is always a Cauchy sequence, but a Cauchy sequence whose elements belong to a field F does not necessarily converge to an element in F. To give an example, let us suppose that F is the set of rational numbers; consider the sequence of integers defined by the Fibonacci recurrence relation:

It can be shown that:

      (1.14)

      Now define the sequence of rational numbers by:

      We can show that:

      and this limit is not a rational number. The Fibonacci numbers are useful in many kinds of applications, such as optimisation (finding the minimum or maximum of a function) and random number generation.

      We define a complete metric space X as one in which every Cauchy sequence converges to an element in X. Examples of complete metric spaces are:

       Euclidean space .

       The metric space C[a, b] of continuous functions on the interval [a, b].

       By definition, Banach spaces are complete normed linear spaces. A normed linear space has a norm based on a metric, as follows .

        is the Banach space of functions defined by the norm for .

      Definition 1.6 An open cover of a set E in a metric space X is a collection of open subsets of X such that .

      Finally, we say that a subset K of a metric space X is compact if every open cover of K contains a finite subcover, that is for some finite N.

      1.6.3 Lipschitz Continuous Functions

      We now examine functions that map one metric space into another one. In particular, we discuss the concepts of continuity and Lipschitz continuity.

      It is convenient to discuss these concepts in the context of metric spaces.

      Definition 1.7 Let and be two metric spaces. A function f from X into Y is said to be continuous at the point if for each there exists a such that:

This is a generalisation of the concept of continuity in Section 1.2 (Definition 1.1). We should note that this definition refers to the continuity of a function at a single point. Thus, a function can be continuous at some points and discontinuous at other points.

      Definition 1.8 A function f from a metric space into a metric space is said to be a uniformly continuous on a set if for each there exists a such that:

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