# Combination Theorem for Sequences/Real/Quotient Rule

## Theorem

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:

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

provided that $m \ne 0$.

### Corollary

Let $\sequence {x_n}$ be convergent to the following limit:

- $\displaystyle \lim_{n \mathop \to \infty} x_n = l$

Then:

- $\displaystyle \lim_{n \mathop \to \infty} \frac 1 {x_n} = \frac 1 l$

provided that $l \ne 0$.

## Proof

As $y_n \to m$ as $n \to \infty$, it follows from Modulus of Limit that $\size {y_n} \to \size m$ as $n \to \infty$.

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

As the statement is given, it is possible that $y_n = 0$ for some $n$.

At such $n$, the terms $\dfrac {x_n} {y_n}$ are not defined.

However, from Sequence Converges to Within Half Limit, we have:

- $\exists N: \forall n > N: \size {y_n} > \dfrac {\size m} 2$

Hence for all $n > N$ we have that $y_n \ne 0$.

Thus we may restrict our attention to the domain of $\sequence {y_n}$ such that $n > N$, knowing that $\dfrac {x_n} {y_n}$ will be defined in that domain.

So, for $n > N$, consider:

\(\displaystyle \size {\frac {x_n} {y_n} - \frac l m}\) | \(=\) | \(\displaystyle \size {\frac {m x_n - y_n l} {m y_n} }\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle \frac 2 {\size m^2} \size {m x_n - y_n l}\) |

By the above, $m x_n - y_n l \to m l - m l = 0$ as $n \to \infty$.

The result follows by the Squeeze Theorem for Real 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

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.8 \ \text{(iii)}$: Criteria for convergence - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): Appendix: $\S 18.3$: Combination theorem - 1953: Walter Rudin:
*Principles of Mathematical Analysis*: $3.3 \ \text{d}$