Cauchy Sequence is Bounded/Normed Division Ring/Proof 1

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

Every Cauchy sequence in $R$ is bounded.


Let $\sequence {x_n} $ be a Cauchy sequence in $R$ then:

$\forall \epsilon \in \R_{\gt 0}: \exists N \in \N : \forall n, m \ge N: \norm {x_n - x_m} \lt \epsilon$.

Let $n_1$ satisfy:

$\forall n, m \ge n_1: \norm {x_n - x_m} \lt 1$

Then $\forall n \ge n_1$:

\(\displaystyle \norm {x_n}\) \(=\) \(\displaystyle \norm {x_n - x_{n_1} + x_{n_1} }\) $\quad$ $\quad$
\(\displaystyle \) \(\le\) \(\displaystyle \norm {x_n - x_{n_1} } + \norm {x_{n_1} }\) $\quad$ Norm axiom (N3) (Triangle Inequality). $\quad$
\(\displaystyle \) \(\le\) \(\displaystyle 1 + \norm {x_{n_1} }\) $\quad$ Since $n, n_1 \ge n_1$. $\quad$

Let $K = \max \set {\norm {x_1}, \norm {x_2}, \dots, \norm {x_{n_1 - 1}}, 1 + \norm {x_{n_1} } }$ then:

$\forall n \lt n_1: \norm {x_n} \le K$
$\forall n \ge n_1: \norm {x_n} \le 1 + \norm {x_{n_1} } \le K$

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