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 $m \ne 0$.

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:

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