# Definition:Limit of Sequence (Number Field)

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## Definition

As:

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

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

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

### Rational Numbers

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

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

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

### Complex Numbers

Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$.

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

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