Difference between revisions of "Convergent Sequences in Metric Spaces"

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* [http://mathonline.wikidot.com/limits-of-sequences-in-metric-spaces Limits of Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
 
* [http://mathonline.wikidot.com/limits-of-sequences-in-metric-spaces Limits of Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
 
* [http://mathonline.wikidot.com/the-boundedness-of-convergent-sequences-in-metric-spaces The Boundedness of Convergent Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
 
* [http://mathonline.wikidot.com/the-boundedness-of-convergent-sequences-in-metric-spaces The Boundedness of Convergent Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
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* [http://mathonline.wikidot.com/convergent-sequences-and-subsequences-in-metric-spaces Convergent Sequences and Subsequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

Latest revision as of 11:00, 8 November 2021

Limits of Sequences in Metric Spaces

Recall that if a sequence of real numbers is an infinite ordered list where for every . We will now generalize the concept of a sequence to contain elements from a metric space .

Definition: Let be a metric space. An (infinite) Sequence in denoted is an infinite ordered list of elements for all .

Finite sequences in a metric space can be defined as a finite ordered list of elements in but their study is not that interesting to us.

We can also define whether a sequence of elements from a metric space converges or diverges.

Definition: Let be a metric space. A sequence in is said to be Convergent to the element written if and the element is said to be the Limit of the sequence . If no such exists, then is said to be Divergent.

There is a subtle but important point to make. In the definition above, represents the limit of a sequence of elements from the metric space to an element while represents the limit of a sequence of positive real numbers to - such limits we already have experience with.

Convergent sequence in metric space

For example, if is any nonempty set, is the discrete metric, and , then the sequence defined by for all , then the sequence:

Furthermore, it's not hard to see that this sequence converges to , i.e., , i.e., since for all we have that , so .

We will soon see that many of theorems regarding limits of sequences of real numbers are analogous to limits of sequences of elements from metric spaces.

The Boundedness of Convergent Sequences in Metric Spaces

If is a metric space and is a sequence in that is convergent then the limit of this sequence is unique.

We will now look at another rather nice theorem which states that if is convergent then it is also bounded.

Theorem 1: Let be a metric space and let be a sequence in . If is convergent then the set is bounded.

  • Proof: Let be a metric space and let be a sequence in that converges to , i.e., . Then . So for all there exists an such that if then . So for there exists an such that if then:
  • Now consider the elements . This is a finite set of elements and furthermore the set of distances from these elements to is finite:
  • Define to be the maximum of these distances:
  • So if we have that and if then . Let . Then for all , . So consider the open ball . Then for all so:
  • Therefore is a bounded set in .

Convergent Sequences and Subsequences in Metric Spaces

We will now look at some nice criterion which tells us that in a metric space , a sequence converges to if and only if every subsequence converges to . This is analogous to the similar result when we looked at convergent sequences of real numbers.

Theorem 1: Let be a metric space, let be a sequence in , and let . Then converges to if and only if every subsequence of converges to .

  • Proof: Let be a convergent sequence in that converges to . Then . So . Therefore, for all there exists an such that if then:
  • Let be any subsequence of . For each we have that for some that and so for all we have that so by the convergence of we have that:
  • So for each there exists a such that if then , therefore, the subsequence converges to .
  • Suppose that every subsequence of converges to . Then is a subsequence of itself and converges to .

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