Combination Theorem for Cauchy Sequences/Constant Rule
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Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Let $a \in R$.
The constant sequence $\tuple {a, a, a, \dots}$ is a Cauchy sequence.
Proof
Let $\sequence {x_n}$ be the constant sequence:
- $\forall n, x_n = a$
Given $\epsilon > 0$:
- $\forall n, m \ge 1: \norm {x_n - x_m} = \norm {a - a} = \norm {0} = 0 < \epsilon$
The result follows.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions