Null Sequences form Maximal Left and Right Ideal/Lemma 6

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:

$\quad \forall \sequence {x_n} \in \NN, \sequence {y_n} \in \CC: \sequence {x_n} \sequence {y_n} \in \NN, \sequence {y_n} \sequence {x_n} \in \NN$

Proof

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

By the definition of the product on the ring of Cauchy sequences then:

$\sequence {x_n} \sequence {y_n} = \sequence {x_n y_n}$

$\sequence {y_n} \sequence {x_n} = \sequence {y_n x_n}$

By Product of Sequence Converges to Zero with Cauchy Sequence Converges to Zero:

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

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

The result follows.

$\blacksquare$

Sources

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