Null Sequences form Maximal Left and Right Ideal/Lemma 6

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

Let $\mathcal {C}$ be the ring of Cauchy sequences over $R$

Let $\mathcal {N}$ be the set of null sequences.

Then:
 * $\quad \forall \sequence {x_n} \in \mathcal {N}, \sequence {y_n} \in \mathcal {C}: \sequence {x_n} \sequence {y_n} \in \mathcal {N}, \sequence {y_n} \sequence {x_n} \in \mathcal {N}$

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

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 then:
 * $\displaystyle \lim_{n \mathop \to \infty} x_n y_n = 0$
 * $\displaystyle \lim_{n \mathop \to \infty} y_n x_n = 0$

The result follows.