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:

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

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

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

Then the following results hold:

Sum Rule

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

Difference Rule

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

Multiple Rule

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

Combined Sum Rule

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

Product Rule

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

Quotient Rule

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

provided that $d \ne 0$.