Definition:Convergent Sequence/Complex Numbers/Definition 1

Definition

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

$\sequence {z_k}$ converges to the limit $c \in \C$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$

where $\cmod z$ denotes the modulus of $z$.

Note on Domain of $N$

Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.

Geometrical Interpretation

The following illustration is a geometrical interpretation of the concept of a Convergent Complex Sequence.

As $n$ increases, the distance from $z_n$ to $c$ gets smaller, until after $n$ is greater than $N$, $\cmod {z_n - c}$ is always less than $\epsilon$:

Also see

• Equivalence of Definitions of Convergent Complex Sequence

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Definition.