Combination Theorem for Cauchy Sequences/Combined Sum Rule

Theorem

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

Let $\sequence {x_n}$, $\sequence {y_n}$ be Cauchy sequences in $R$.

Let $a, b \in R$.

Then:

$\sequence {a x_n + b y_n }$ is a Cauchy sequence.

Proof

From the Multiple Rule for Normed Division Ring Sequences:

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

$\sequence {b y_n}$ is a Cauchy sequence.

The result now follows directly from the Sum Rule for Normed Division Ring Sequences:

$\sequence {a x_n + b y_n}$ is a Cauchy sequence.

$\blacksquare$

Sources

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