Combination Theorem for Cauchy Sequences/Difference Rule

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

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

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

Proof
From Multiple Rule for Normed Division Ring Sequences:
 * $\sequence {-y_n} = \sequence {\paren {-1} y_n} $ is a Cauchy sequence.

From Sum Rule for Normed Division Ring Sequences:
 * $\sequence {x_n - y_n} = \sequence {x_n + \paren {-y_n}}$ is a Cauchy sequence.