Definition:Limit of Sequence

Definition

Topological Space

Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

Let $\sequence {x_n}$ be a sequence in $A$.

Let $\sequence {x_n}$ converge to a value $\alpha \in S$.

Then $\alpha$ is known as a limit (point) of $\sequence {x_n}$ (as $n$ tends to infinity).

Metric Space

Let $M = \struct {A, d}$ be a metric space or pseudometric space.

Let $\sequence {x_n}$ be a sequence in $M$.

Let $\sequence {x_n}$ converge to a value $l \in A$.

Then $l$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity.

If $M$ is a metric space, this is usually written:

$\ds l = \lim_{n \mathop \to \infty} x_n$

Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\sequence {x_n} $ be a sequence in $R$.

Let $\sequence {x_n}$ converge to $x \in R$.

Then $x$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity which is usually written:

$\ds x = \lim_{n \mathop \to \infty} x_n$

Normed Vector Space

Let $L \in X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ converge to $L$.

Then $L$ is a limit of $\sequence {x_n}_{n \mathop \in \N}$ as $n$ tends to infinity which is usually written:

$\ds L = \lim_{n \mathop \to \infty} x_n$

Test Function Space

Let $\map \DD {\R^d}$ be the test function space.

Let $\phi \in \map \DD {\R^d}$ be a test function.

Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence of test functions in $\map \DD {\R^d}$.

Let $\sequence {\phi_n}_{n \mathop \in \N}$ converge to $\phi$ in $\map \DD {\R^d}$.

Then $\phi$ is a limit of $\sequence {\phi_n}_{n \mathop \in \N}$ in $\map \DD {\R^d}$ as $n$ tends to infinity which is usually written:

$\phi_n \stackrel \DD {\longrightarrow} \phi$

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 $\C$:

Real Numbers

Let $\sequence {x_n}$ be a sequence in $\R$.

Let $\sequence {x_n}$ converge to a value $l \in \R$.

Then $l$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity.

Rational Numbers

Let $\sequence {x_n}$ be a sequence in $\Q$.

Let $\sequence {x_n}$ converge to a value $l \in \R$, where $\R$ denotes the set of real numbers.

Then $l$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity.

Complex Numbers

Let $\sequence {z_n}$ be a sequence in $\C$.

Let $\sequence {z_n}$ converge to a value $l \in \C$.

Then $l$ is a limit of $\sequence {z_n}$ as $n$ tends to infinity.

$p$-adic Numbers

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\, \cdot \,}_p}$ be the $p$-adic numbers.

Let $\sequence {x_n} $ be a sequence in $\Q_p$.

Let $\sequence {x_n}$ converge to $x \in \Q_p$

Then $x$ is a limit of $\sequence {x_n}$ as $n$ tends to infinity which is usually written:

$\ds x = \lim_{n \mathop \to \infty} x_n$

Also see

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

• Definition:Limit of Mapping

• Results about limits of sequences can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): tend to
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): limit (of a sequence)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): tend to