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 $a \in R$.

Then:

$\sequence {a x_n}$ is a Cauchy sequence.

Proof

Follows directly from Product Rule for Normed Division Ring Sequences, setting

$\sequence {y_n} := \sequence {x_n}$

and:

$\sequence {x_n} := \tuple {a, a, a, \ldots}$

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions