Combination Theorem for Cauchy Sequences/Multiple Rule

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

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

Let $\lambda \in R$.

Then:
 * $\sequence {\lambda x_n}$ is a Cauchy sequence.

Proof
Follows directly from Product Rule for Normed Division Ring Sequences, setting
 * $\sequence {y_n} := \sequence {x_n}$

and:
 * $\sequence {x_n} := \tuple {\lambda, \lambda, \lambda, \ldots}$