Combination Theorem for Limits of Functions/Complex/Multiple Rule

Theorem

Let $\C$ denote the complex numbers.

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

Let $f$ tend to the following limit:

$\ds \lim_{z \mathop \to c} \map f z = l$

Let $\lambda \in \C$ be an arbitrary complex number.

Then:

$\ds \lim_{z \mathop \to c} \lambda \map f z = \lambda l$

Proof

Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:

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

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

By Limit of Complex Function by Convergent Sequences:

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

By the Multiple Rule for Complex Sequences:

$\ds \lim_{n \mathop \to \infty}\lambda \map f {z_n} = \lambda l$

Applying Limit of Complex Function by Convergent Sequences again:

$\ds \lim_{z \mathop \to c} \lambda \map f z = \lambda l$

$\blacksquare$