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 |
$\blacksquare$