Eventually Constant Sequence Converges to Constant

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

Proof
Let $\sequence {y_n}$ be the subsequence of $\sequence {\norm {x_n}}$ defined by:
 * $\forall n: y_n =x_{N + n}$

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

Then: