Null Sequences form Maximal Left and Right Ideal/Lemma 7

Theorem

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

Let $\CC$ be the ring of Cauchy sequences over $R$

Let $\NN$ be the set of null sequences.

Then:

$\NN \subsetneq \CC$.

Proof

By every convergent sequence is a Cauchy sequence then $\NN \subseteq \CC$.

From Constant Sequence Converges to Constant in Normed Division Ring, the unity $\tuple {1, 1, 1, \dotsc}$ of $\CC$ converges to $1 \in R$, and therefore $\tuple {1, 1, 1, \dotsc} \in \CC \setminus \NN$

So $\NN \subsetneq \CC$.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions