Convergent Sequence in Normed Division Ring is Bounded/Proof 1

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

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

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:
 * $\forall n < n_1: \norm {x_n} \le K$
 * $\forall n \ge n_1: \norm {x_n} \le 1 + \norm l \le K$

Thus, by definition, $\sequence {x_n}$ is bounded.