Combination Theorem for Sequences/Quotient Rule

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Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be sequences in $X$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be convergent to the following limits:

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

Then:

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

provided that $m \ne 0$.

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

As $m \ne 0$, it follows from the definition of the modulus of $m$ that $\left\vert{m}\right\vert > 0$.

From Sequence Converges to Within Half Limit, we have $\exists N: \forall n > N: \left\vert{y_n}\right\vert > \frac {\left\vert{m}\right\vert} 2$.

Now, for $n > N$, consider:

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left\vert \frac {x_n} {y_n} - \frac {l} {m} \right\vert$	$=$	$\displaystyle \left\vert \frac {m x_n - y_n l} {m y_n} \right\vert$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$<$	$\displaystyle \frac 2 {\left\vert{m}\right\vert^2} \left\vert{m x_n - y_n l}\right\vert$	$\displaystyle $	$\displaystyle $	$\displaystyle $

By the above, $m x_n - y_n l \to ml - ml = 0$ 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).

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left\vert \frac {x_n} {y_n} - \frac {l} {m} \right\vert\)	\(=\)	\(\displaystyle \left\vert \frac {m x_n - y_n l} {m y_n} \right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(<\)	\(\displaystyle \frac 2 {\left\vert{m}\right\vert^2} \left\vert{m x_n - y_n l}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)