Combination Theorem for Sequences/Multiple Rule

Theorem

Let $\sequence {x_n}$ be a sequences in $\R$.

Let $\sequence {x_n}$ be convergent to the following limit:

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

Let $\lambda \in \R$.

Then:

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

Let $\sequence {z_n}$ be a sequence in $\C$.

Let $\sequence {z_n}$ be convergent to the following limit:

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

Let $\lambda \in \C$.

Then:

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

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): convergent series
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convergent series