Combination Theorem for Sequences/Product 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:

$\displaystyle\lim_{n \to \infty} \left({x_n y_n}\right) = l m$

Since $\left \langle {x_n} \right \rangle$ converges, it is bounded by Convergent Sequence is Bounded.

Suppose $\left\vert{x_n}\right\vert \le K$ for $n = 1, 2, 3, \ldots$.

Then:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left\vert{x_n y_n - l m}\right\vert$	$=$	$\displaystyle \left\vert{x_n y_n - x_n m + x_n m - l m}\right\vert$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\le$	$\displaystyle \left\vert{x_n y_n - x_n m}\right\vert + \left\vert{x_n m - l m}\right\vert$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Triangle Inequality
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\le$	$\displaystyle \left\vert{x_n}\right\vert \cdot \left\vert{y_n - m}\right\vert + \left\vert{m}\right\vert \cdot \left\vert{x_n - l}\right\vert$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Modulus of Product
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\le$	$\displaystyle K \cdot \left\vert{y_n - m}\right\vert + \left\vert{m}\right\vert \cdot \left\vert{x_n - l}\right\vert$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle z_n$	$\displaystyle $	$\displaystyle $	$\displaystyle $

But $x_n \to l$ as $n \to \infty$.

So $\left\vert{x_n - l}\right\vert \to 0$ as $n \to \infty$ from Convergent Sequence Minus Limit.

Similarly $\left\vert{y_n - m}\right\vert \to 0$ as $n \to \infty$.

$\lim_{n \to \infty} \left({\lambda x_n + \mu y_n}\right) = \lambda l + \mu m$, $z_n \to 0$ 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).

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left\vert{x_n y_n - l m}\right\vert\)	\(=\)	\(\displaystyle \left\vert{x_n y_n - x_n m + x_n m - l m}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\le\)	\(\displaystyle \left\vert{x_n y_n - x_n m}\right\vert + \left\vert{x_n m - l m}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Triangle Inequality
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\le\)	\(\displaystyle \left\vert{x_n}\right\vert \cdot \left\vert{y_n - m}\right\vert + \left\vert{m}\right\vert \cdot \left\vert{x_n - l}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Modulus of Product
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\le\)	\(\displaystyle K \cdot \left\vert{y_n - m}\right\vert + \left\vert{m}\right\vert \cdot \left\vert{x_n - l}\right\vert\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle z_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)