Convergent Sequence in Normed Division Ring is Bounded/Proof 1

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

Let $\sequence {x_n}$ be a sequence in $R$.

Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the following limit:


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

Then $\sequence {x_n}$ is bounded.

Proof
Let $\sequence {x_n}$ be convergent in the norm $\norm {\,\cdot\,}$ to the limit $l$, then:
 * $\forall \epsilon \in \R_{\gt 0}: \exists N \in \N : \forall n \ge N: \norm {x_n - l} \lt \epsilon$.

Let $n_1$ satisfy:
 * $\forall n \ge n_1: \norm {x_n - l} \lt 1$

Then $\forall n \ge n_1$:

Let $K = \max \set {\norm {x_1}, \norm {x_2}, \dots, \norm {x_{n_1 - 1}}, 1 + \norm {l} }$ then:
 * $\forall n \lt n_1: \norm {x_n} \le K$
 * $\forall n \ge n_1: \norm {x_n} \le 1 + \norm {l } \le K$

By the definition of a bounded sequence in a normed division ring then $\sequence {x_n}$ is bounded.