Combination Theorem for Sequences/Complex

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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 $d \ne 0$.