Combination Theorem for Sequences/Normed Division Ring/Constant Rule

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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 $\displaystyle \lim_{n \mathop \to \infty} \lambda = \lambda$


Proof

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

$\forall n, x_n = \lambda$

Given $\epsilon \gt 0$, then:

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

The result follows.


$\blacksquare$

Sources