Null Sequences form Maximal Left and Right Ideal/Lemma 5

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:

$\forall \sequence {x_n}, \sequence {y_n} \in \NN: \sequence {x_n} + \paren {-\sequence {y_n} } \in \NN$

Proof

Let $\ds \lim_{n \mathop \to \infty} x_n = 0$ and $\ds \lim_{n \mathop \to \infty} y_n = 0$.

The sequence $\sequence {x_n} + \paren {-\sequence {y_n} } = \sequence {x_n - y_n}$.

From Difference Rule for Sequences in Normed Division Ring:

$\ds \lim_{n \mathop \to \infty} x_n - y_n = 0 - 0 = 0$

The result follows.

$\blacksquare$

Sources

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