Limit of Function by Convergent Sequences/Complex Plane

Theorem
Let $f$ be a complex function defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.

Then $\ds \lim_{x \mathop \to c} \map f z = l$ :
 * for each sequence $\sequence {z_n}$ of points of $S$ such that $\forall n \in \N_{>0}: z_n \ne c$ and $\ds \lim_{n \mathop \to \infty} z_n = c$

it is true that:
 * $\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$

Necessary Condition
Let $\ds \lim_{z \mathop \to c} \map f z = l$.

Let $\epsilon \in \R_{>0}$.

Then by definition of the limit of a complex function:
 * $\exists \delta \in \R_{>0}: \cmod {\map f z - l} < \epsilon$

provided $0 < \cmod {z - c} < \delta$.

Now suppose that $\sequence {x_n}$ is a sequence of elements of $S$ such that:
 * $\forall n \in \N_{>0}: z_n \ne c$

and:
 * $\ds \lim_{n \mathop \to \infty} z_n = c$

Since $\delta > 0$, from the definition of the limit of a complex function:
 * $\exists N \in \R_{>0}: \forall n \in \N: n > N \implies \cmod {z_n - c} < \delta$

But:
 * $\forall n \in \N_{>0}: z_n \ne c$

That means:
 * $0 < \cmod {z_n - c} < \delta$

But that implies:
 * $\cmod {\map f {z_n} - l} < \epsilon$

That is, given a value of $\epsilon > 0$, we have found a value of $N$ such that:
 * $\forall n \in \N: n > N \implies \cmod {\map f {z_n} - l} < \epsilon$

Thus:
 * $\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$

Sufficient Condition
Suppose that for each sequence $\sequence {x_n}$ of elements of $S$ such that:
 * $\forall n \in \N_{>0}: z_n \ne c$ and $\ds \lim_{n \mathop \to \infty} z_n = c$, it is true that:
 * $\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$

it is not true that:
 * $\ds \lim_{z \mathop \to c} \map f z = l$

Thus:
 * $\exists \epsilon > 0: \forall \delta > 0: \exists x: 0 < \cmod {z - c} < \delta: \cmod {\map f {z_n} - l} \ge \epsilon$

In particular, if $\delta = \dfrac 1 n$, we can find a $z_n$ where $0 < \cmod {x - c} < \dfrac 1 n$ such that:
 * $\cmod {\map f {z_n} - l} \ge \epsilon$

But then $\sequence {z_n}$ is a sequence of elements of $S$ such that:
 * $\forall n \in \N_{>0}: z_n \ne c$ and $\ds \lim_{n \mathop \to \infty} z_n = c$

but for which it is not true that:
 * $\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$

The result follows by Proof by Contradiction.