Combination Theorem for Cauchy Sequences/Sum Rule

Theorem
Let $\left({R, \left\Vert{\,\cdot\,}\right\Vert_R}\right)$ be a normed division ring.

Let $\left({V, \left\Vert{\,\cdot\,}\right\Vert}\right)$ be a normed vector space over $R$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be Cauchy sequences in $V$.

Then:
 * $\left \langle{x_n + y_n}\right \rangle$ is a Cauchy sequence in $V$.

Proof
Let $\epsilon > 0$ be given.

Then $\dfrac \epsilon 2 > 0$.

Since $\left \langle {x_n} \right \rangle$ is a Cauchy sequence, we can find $N_1$ such that:
 * $\forall n, m > N_1: \left\Vert{x_n - x_m}\right\Vert < \dfrac \epsilon 2$

Similarly, $\left \langle {y_n} \right \rangle$ is a Cauchy sequence, we can find $N_2$ such that:
 * $\forall n, m > N_2: \left\Vert{y_n - y_m}\right\Vert < \dfrac \epsilon 2$

Now let $N = \max \left\{{N_1, N_2}\right\}$.

Then if $n, m > N$, both the above inequalities will be true.

Thus $\forall n, m > N$:

Hence:
 * $\left \langle{x_n + y_n}\right \rangle$ is a Cauchy sequence in $V$.