Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-18T18:40:56.825Z Has data issue: false hasContentIssue false

3 - Sequences in Metric Spaces

Published online by Cambridge University Press:  05 April 2012

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 〈xn〉 converging to x implies that 〈f(xn)〉 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 〈xn〉 be a sequence in X. Then 〈Xn〉 is said to converge to a point xX if for each real number ε>0, there exists an m∈ℤ+ such that d(xn, x) <ε for all nm.

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

Type
Chapter
Information
Publisher: Foundation Books
Print publication year: 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

• Sequences in Metric Spaces
• Book: First Course in Metric Spaces
• Online publication: 05 April 2012
• Chapter DOI: https://doi.org/10.1017/UPO9788175968608.004
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

• Sequences in Metric Spaces
• Book: First Course in Metric Spaces
• Online publication: 05 April 2012
• Chapter DOI: https://doi.org/10.1017/UPO9788175968608.004
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

• Sequences in Metric Spaces
• Book: First Course in Metric Spaces
• Online publication: 05 April 2012
• Chapter DOI: https://doi.org/10.1017/UPO9788175968608.004
Available formats
×