Definition:Convergent Sequence (Analysis)

Definition

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $\left \langle {x_k} \right \rangle$ be a sequence in $X$.

$\left \langle {x_k} \right \rangle$ converges to the limit $l$ iff:

$\forall \epsilon > 0: \exists N \in X: n > N \implies \left|{x_n - l}\right| < \epsilon$

where $\left|{x}\right|$ is the modulus of $x$.

The validity of this definition derives from the fact that:

• the Rational Numbers form Metric Space
• the Real Number Line is Metric Space
• the Complex Plane is Metric Space.