Cauchy Sequence is Bounded/Normed Division Ring/Proof 3

Theorem

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

Every Cauchy sequence in $R$ is bounded.

Proof

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

By Norm Sequence of Cauchy Sequence has Limit, $\sequence {\norm {x_n} }$ is a convergent sequence in $\R$.

By Convergent Real Sequence is Bounded, $\sequence {\norm {x_n} }$ is bounded.

That is:

$\exists M \in \R_{\gt 0}: \forall n \in \N: \norm {x_n} = \size {\norm {x_n} } \le M$

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

$\blacksquare$

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.2$: Normed Fields, Exercise $11 \ (2)$