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3 - Sequences in Metric Spaces

Published online by Cambridge University Press:  05 April 2012

B. K. Tyagi
Affiliation:
Associate Professor, Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi
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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.

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Publisher: Foundation Books
Print publication year: 2010

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  • Sequences in Metric Spaces
  • B. K. Tyagi, Associate Professor, Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi
  • Book: First Course in Metric Spaces
  • Online publication: 05 April 2012
  • Chapter DOI: https://doi.org/10.1017/UPO9788175968608.004
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  • Sequences in Metric Spaces
  • B. K. Tyagi, Associate Professor, Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi
  • Book: First Course in Metric Spaces
  • Online publication: 05 April 2012
  • Chapter DOI: https://doi.org/10.1017/UPO9788175968608.004
Available formats
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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
  • B. K. Tyagi, Associate Professor, Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi
  • Book: First Course in Metric Spaces
  • Online publication: 05 April 2012
  • Chapter DOI: https://doi.org/10.1017/UPO9788175968608.004
Available formats
×