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.

$\blacksquare$

Sources

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