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