# 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:

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

Then:

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

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:

\(\ds \size {\frac {z_n} {w_n} - \frac c d}\) | \(=\) | \(\ds \size {\frac {d z_n - w_n c} {d w_n} }\) | ||||||||||||

\(\ds \) | \(<\) | \(\ds \frac 2 {\size m^2} \size {d z_n - w_n c}\) |

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 Complex Sequences (which applies as well to real as to complex sequences).

$\blacksquare$

## Also presented as

Some treatments of this subject specifically exclude all sequences where the denominators are zero at **any** point in their domain.

Thus, for example, this is how it is presented in 1960: Walter Ledermann: *Complex Numbers*:

*If $z_n \to c$ and $w_m \to d$, then**... $\text{(iv)} \ z_n / w_n \to c / d$, where ... $w_n \ne 0$ for all $n$ and $d \ne 0$.*

However, it is demonstrated within the proof that past a certain $N \in \R$, which is bound to exist, $w_n$ is *guaranteed* to be non-zero.

The behaviour of the sequence $S = \sequence {\dfrac {z_n} {w_n} }$ in the limit is not dependent upon the existence or otherwise of its terms for $n < N$.

Thus it is not necessary to state that $w_n \ne 0$ for all $n$, and in fact such a statement would unnecessarily restrict the applicability of the theorem to exclude otherwise well-behaved cases where it is desirable that the theorem *does* apply.

Hence this restriction is not supported on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.2$. Sequences: Rules. $\text {(iv)}$