Constant Sequence Converges to Constant in Normed Division Ring
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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 $\ds \lim_{n \mathop \to \infty} \lambda = \lambda$
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.
$\blacksquare$