Combination Theorem for Cauchy Sequences/Constant Rule

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

Let $\lambda \in R$.

Then:
 * the constant sequence $\tuple {\lambda, \lambda, \lambda, \dots}$ is a Cauchy sequence.

Proof
Let $\sequence {x_n}$ be the constant sequence:
 * $\forall n, x_n = \lambda$

Given $\epsilon \gt 0$, then:
 * $\forall n,m \ge 1: \norm {x_n - x_m} = \norm {\lambda - \lambda} = \norm {0} = 0 < \epsilon$

The result follows.