Combination Theorem for Cauchy Sequences/Sum Rule

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

Let $\sequence {x_n}$ and $\sequence {y_n}$ be Cauchy sequences in $R$.

Then:
 * $\sequence {x_n + y_n}$ is a Cauchy sequence.

Proof
Let $\epsilon > 0$ be given.

Then $\dfrac \epsilon 2 > 0$.

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

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

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

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

Thus $\forall n, m > N$:

Hence:
 * $\sequence {x_n + y_n}$ is a Cauchy sequences in $R$.