Combination Theorem for Sequences/Quotient Rule

Quotient Rule for Real Sequences
Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:


 * $\displaystyle \lim_{n \mathop \to \infty} x_n = l$
 * $\displaystyle \lim_{n \mathop \to \infty} y_n = m$

Then:

Quotient Rule for Complex Sequences
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:


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

Then: