Definition:Limit of a Sequence

Topological Space

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

Let $A \subseteq S$.

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

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

Then $\alpha$ is known as a limit (point) 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:

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

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.

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 of the limit of a sequence in a metric space holds for sequences in the standard number fields $\Q$, $\R$ and $\C$.

Also see

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

• Limit of a Function.