Constant Sequence Converges to Constant in Normed Division Ring

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 convergent and $\ds \lim_{n \mathop \to \infty} \lambda = \lambda$

Proof

Let $\sequence {x_n}$ be the constant sequence:

$\forall n \in \N: x_n = \lambda$

Given $\epsilon \in \R_{>0}$:

$\forall n \ge 1: \norm {x_n - \lambda} = \norm {\lambda - \lambda} = \norm 0 = 0 < \epsilon$

The result follows.

$\blacksquare$