Definition:Limit of Sequence

Topological Space
Let $$T = \left({A, \vartheta}\right)$$ be a topological space.

Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $$\left({A, \vartheta}\right)$$.

Let $$\left \langle {x_n} \right \rangle$$ converge to a value $$l \in A$$.

Then $$l$$ is known as a limit of $$\left \langle {x_n} \right \rangle$$ as $$n$$ tends to infinity.

Metric Space
Let $$\left({X, d}\right)$$ be a metric space.

Let $$\left \langle {x_n} \right \rangle$$ be a sequence in $$\left({X, d}\right)$$.

Let $$\left \langle {x_n} \right \rangle$$ converge to a value $$l \in X$$.

Then $$l$$ is known as the limit of $$\left \langle {x_n} \right \rangle$$ as $$n$$ tends to infinity and is usually written:


 * $$l = \lim_{n \to \infty} x_n$$

It can be seen that by the definition of open set in a metric space, this definition is equivalent to that for a limit in a topological space.

From Sequence in Metric Space has One Limit at Most, it follows that the limit, if it exists, is unique.

Standard Number Fields
As:
 * The set of rational numbers $$\Q$$ under the usual metric forms a metric space;
 * The real number line $$\R$$ under the usual metric forms a metric space;
 * The complex plane $$\C$$ under the usual metric forms a metric space;

the definition holds for sequences in $$\Q$$, $$\R$$ and $$\C$$.

Also see

 * Limit of Sets for an extension of this concept into the field of measure theory.