Combination Theorem for Cauchy Sequences/Difference Rule
< Combination Theorem for Cauchy Sequences(Redirected from Difference Rule for Cauchy Sequences in Normed Division Ring)
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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