Combination Theorem for Sequences/Normed Division Ring/Multiple Rule

Theorem

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

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

Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:

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

Let $\lambda \in R$.

Then:

$\sequence {\lambda x_n}$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$

Proof

Let

$\sequence {\tilde{x}_n} := \tuple {\lambda, \lambda, \lambda, \ldots}$

and:

$\sequence {y_n} := \sequence {x_n}$

The claim follows from Product Rule for Sequences in Normed Division Ring, since:

$\sequence {\lambda x_n} = \sequence {\tilde{x}_n y_n}$

$\blacksquare$