Combination Theorem for Sequences/Product 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:

$\ds \lim_{n \mathop \to \infty} y_n$

$=$

$\ds l$

$\ds \lim_{n \mathop \to \infty} x_n$

$=$

$\ds m$

Then:

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

Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.

Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:

$\ds \lim_{n \mathop \to \infty} z_n$

$=$

$\ds c$

$\ds \lim_{n \mathop \to \infty} w_n$

$=$

$\ds d$

Then:

$\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$

2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebra of limits