Combination Theorem for Cauchy Sequences/Constant Rule

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

Let $a \in R$.

Then:
 * 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.