Convergent Sequences in Metric Spaces

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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.

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