Null Sequences form Maximal Left and Right Ideal/Lemma 4

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:

$\NN \ne \O$

Proof

From Constant Sequence Converges to Constant in Normed Division Ring, the zero $\tuple {0, 0, 0, \dots}$ of $\CC$ to converges $0 \in R$.

Therefore $\tuple {0, 0, 0, \dots} \in \NN$.

$\blacksquare$

Sources

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