# Combination Theorem for Sequences/Complex/Combined Sum Rule

## Theorem

Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.

Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:

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

Let $\lambda, \mu \in \C$.

Then:

$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$

## Proof

From the Multiple Rule for Complex Sequences, we have:

$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$
$\ds \lim_{n \mathop \to \infty} \paren {\mu w_n} = \mu d$

The result now follows directly from the Sum Rule for Complex Sequences:

$\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$

$\blacksquare$