Null Sequences form Maximal Left and Right Ideal/Lemma 5

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:
 * $\forall \sequence {x_n}, \sequence {y_n} \in \mathcal {N}: \sequence {x_n} + \paren {-\sequence {y_n} } \in \mathcal {N}$

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

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

By Difference Rule for Sequences, $\displaystyle \lim_{n \mathop \to \infty} x_n - y_n = 0 - 0 = 0.$

The result follows.