Convergent Sequence in Normed Division Ring is Bounded/Proof 3

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 modulus of limit in normed division ring, $\sequence {\norm {x_n} }$ is a convergent sequence in $\R$.

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

That is:

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

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 $11$ $(1)$