# Combination Theorem for Sequences/Complex

## 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:

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

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

Then the following results hold:

### Sum Rule

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

### Difference Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {z_n - w_n} = c - d$

### Multiple Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda z_n} = \lambda c$

### Combined Sum Rule

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

### Product Rule

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

### Quotient Rule

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

provided that $d \ne 0$.