Combination Theorem for Sequences/Real/Product Rule

From ProofWiki
Jump to navigation Jump to search

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:

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


Proof

Because $\sequence {x_n}$ converges, it is bounded by Convergent Sequence is Bounded.

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

Then:

\(\displaystyle \size {x_n y_n - l m}\) \(=\) \(\displaystyle \size {x_n y_n - x_n m + x_n m - l m}\)
\(\displaystyle \) \(\le\) \(\displaystyle \size {x_n y_n - x_n m} + \size {x_n m - l m}\) Triangle Inequality for Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle \size {x_n} \size {y_n - m} + \size m \size {x_n - l}\) Absolute Value of Product
\(\displaystyle \) \(\le\) \(\displaystyle K \size {y_n - m} + \size m \size {x_n - l}\)
\(\displaystyle \) \(=:\) \(\displaystyle z_n\)


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

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

Similarly $\size {y_n - m} \to 0$ as $n \to \infty$.

From the Combined Sum Rule for Real Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$, $z_n \to 0$ as $n \to \infty$

The result follows by the Squeeze Theorem for Real Sequences.

$\blacksquare$


Sources