Combination Theorem for Sequences/Complex/Quotient 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:


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

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

provided that $d \ne 0$.

Proof
As $z_n \to c$ as $n \to \infty$, it follows from Modulus of Limit that $\size {w_n} \to \size d$ as $n \to \infty$.

As $d \ne 0$, it follows from the definition of the modulus of $d$ that $\size d > 0$.

From Sequence Converges to Within Half Limit, we have $\exists N: \forall n > N: \size {w_n} > \dfrac {\size d} 2$.

Now, for $n > N$, consider:

By the above, $d z_n - w_n c \to d c - d c = 0$ as $n \to \infty$.

The result follows by the Squeeze Theorem for Sequences of Complex Numbers (which applies as well to real as to complex sequences).