Numerical Methods in Computational Finance - Daniel J. Duffy

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      A simple example is to show that the sequence left-brace StartFraction 1 Over n EndFraction right-brace comma n greater-than-or-equal-to 1 converges to 0. To this end, let epsilon be a positive real number. Then there exists a positive integer n 0 greater-than 1 slash epsilon such that bar StartFraction 1 Over n EndFraction minus 0 bar equals StartFraction 1 Over n EndFraction less-than epsilon whenever n greater-than-or-equal-to n 0.

      Definition 1.5 A sequence left-parenthesis a Subscript n Baseline right-parenthesis of elements of an ordered field F is called a Cauchy sequence if for each epsilon greater-than 0 in F there exists a positive integer n 0 such that:

StartAbsoluteValue a Subscript n Baseline minus a Subscript m Baseline EndAbsoluteValue less-than epsilon whenever m comma n greater-than-or-equal-to n 0 period

      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:

StartLayout 1st Row 1st Column upper F 0 2nd Column equals 0 2nd Row 1st Column upper F 1 2nd Column equals 1 3rd Row 1st Column upper F Subscript n 2nd Column equals upper F Subscript n minus 1 Baseline plus upper F Subscript n minus 2 Baseline comma n greater-than-or-equal-to 2 period EndLayout

      (1.14)StartLayout 1st Row 1st Column Blank 2nd Column upper F Subscript n Baseline equals StartFraction 1 Over StartRoot 5 EndRoot EndFraction left-bracket alpha Superscript n Baseline minus beta Superscript n Baseline right-bracket 2nd Row 1st Column Blank 2nd Column where alpha equals StartFraction 1 plus StartRoot 5 EndRoot Over 2 EndFraction beta equals StartFraction 1 minus StartRoot 5 EndRoot Over 2 EndFraction period EndLayout

      Now define the sequence of rational numbers by:

StartLayout 1st Row 1st Column x Subscript n Baseline equals upper F Subscript n Baseline slash upper F Subscript n minus 1 Baseline comma 2nd Column n greater-than-or-equal-to 1 period EndLayout

      We can show that:

limit Underscript n right-arrow infinity Endscripts x Subscript n Baseline equals alpha equals StartFraction 1 plus StartRoot 5 EndRoot Over 2 EndFraction left-parenthesis the italic Golden Ratio right-parenthesis

      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 left-brace upper G Subscript j Baseline right-brace of open subsets of X such that upper E subset-of union upper G Subscript j.

      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 upper K subset-of union upper G Subscript j 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 left-parenthesis upper X comma d 1 right-parenthesis and left-parenthesis upper Y comma d 2 right-parenthesis be two metric spaces. A function f from X into Y is said to be continuous at the point normal a element-of upper X if for each epsilon greater-than 0 there exists a delta greater-than 0 such that:

d 2 left-parenthesis f left-parenthesis x right-parenthesis comma f left-parenthesis a right-parenthesis right-parenthesis less-than epsilon whenever d 1 left-parenthesis x comma a right-parenthesis less-than delta period

      Definition 1.8 A function f from a metric space left-parenthesis upper X comma d 1 right-parenthesis into a metric space left-parenthesis upper Y comma d 2 right-parenthesis is said to be a uniformly continuous on a set upper E subset-of upper X if for each epsilon greater-than 0 there exists a delta greater-than 0 such that:

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