##
*
*Convergent Sequence in a Metric Space

**Definition.**

Let be a sequence of points in the metric space *E*.

A point is called a *limit* of the sequence if,

given any real number ,

there is a positive integer *N* such that whenever .

See ita-3-3-2-page45.
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If the sequence has a limit, we call the sequence *convergent*,

and if *p* is a limit of the sequence, we say that the sequence *converges* to *p*.

See ita-3-3-2-page45.

The statement that the sequence of points (in a metric space *E*)

converges to the point *p* (also in *E*)

is written concisely as

**.**

See ita-3-3-9-page46.