# Combination Theorem for Sequences/Normed Division Ring/Sum Rule

## Theorem

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

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

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limits:

- $\displaystyle \lim_{n \mathop \to \infty} x_n = l$
- $\displaystyle \lim_{n \mathop \to \infty} y_n = m$

Then:

- $\sequence {x_n + y_n}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$

## Proof

Let $\epsilon > 0$ be given.

Then $\dfrac \epsilon 2 > 0$.

Since $\sequence {x_n}$ is convergent to $l$, we can find $N_1$ such that:

- $\forall n > N_1: \norm {x_n - l} < \dfrac \epsilon 2$

Similarly, for $\sequence {y_n}$ we can find $N_2$ such that:

- $\forall n > N_2: \norm {y_n - m} < \dfrac \epsilon 2$

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

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

Thus $\forall n > N$:

\(\displaystyle \norm {\paren {x_n + y_n} - \paren {l + m} }\) | \(=\) | \(\displaystyle \norm {\paren {x_n - l} + \paren {y_n - m} }\) | |||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \norm {x_n - l} + \norm {y_n - m}\) | Axiom (N3) of norm (Triangle Inequality) | ||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle \frac \epsilon 2 + \frac \epsilon 2 = \epsilon\) |

Hence:

- $\sequence {x_n + y_n}$ is convergent to $l + m$.

$\blacksquare$