Eventually Constant Sequence Converges to Constant

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Theorem

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

Let $\lambda \in R$.

Let $\sequence {x_n}$ be a sequence in $R$ such that:

$\exists N \in \R_{>0} : \forall n \ge N: x_n = \lambda$


Then:

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


Proof

Let $\sequence {y_n}$ be the subsequence of $\sequence {\norm {x_n} }$ defined as:

$\forall n: y_n = x_{N + n}$

The $\sequence {y_n}$ is the constant sequence $\tuple {\lambda, \lambda, \lambda, \dotsc}$.


Then:

\(\ds \lim_{n \mathop \to \infty} x_n\) \(=\) \(\ds \lim_{n \mathop \to \infty} y_n\) Limit of Subsequence equals Limit of Sequence in Normed Division Ring
\(\ds \) \(=\) \(\ds \lambda\) Constant Sequence Converges to Constant in Normed Division Ring

$\blacksquare$