Convergent Sequence in Normed Division Ring is Bounded/Proof 4

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:

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

Then $\sequence {x_n}$ is bounded.

Proof

Let $\sequence {x_n}$ be convergent to the limit $l$ in $\struct {R, \norm {\,\cdot\,}}$.

By Convergent Sequence is Cauchy Sequence in Normed Division Ring, $\sequence {x_n}$ is a Cauchy sequence in $\struct {R, \norm {\,\cdot\,}}$.

By Cauchy Sequence in Normed Division Ring is Bounded, $\sequence {x_n}$ is a bounded sequence in $\struct {R, \norm {\,\cdot\,}}$.

$\blacksquare$

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.2$: Normed Fields, Exercise $11$ $(1)$