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 $\lambda, \mu \in R$.

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

Proof
From the Multiple Rule for Normed Division Ring Sequences, we have:
 * $\sequence {\lambda x_n}$ is a Cauchy sequence.
 * $\sequence {\mu y_n}$ is a Cauchy sequence.

The result now follows directly from the Sum Rule for Normed Division Ring Sequences:
 * $\sequence {\lambda x_n + \mu y_n}$ is a Cauchy sequence.