# Combination Theorem for Sequences/Normed Division Ring

## 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:

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

Let $\lambda, \mu \in R$.

Then the following results hold:

### Sum Rule

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

### Difference Rule

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

### Multiple Rule

$\sequence {\lambda x_n}$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$

### Combined Sum Rule

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

### Product Rule

$\sequence {x_n y_n}$ is convergent to the limit $\ds \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$

### Inverse Rule

Suppose $l \ne 0$.

Then:

$\exists k \in \N : \forall n \in \N: x_{k + n} \ne 0$

and the subsequence $\sequence { x_{k+n}^{-1} }$ is well-defined and convergent with:

$\ds \lim_{n \mathop \to \infty} {x_{k + n} }^{-1} = l^{-1}$.

### Quotient Rule

Suppose $m \ne 0$.

Then:

$\exists k \in \N : \forall n \in \N: y_{k + n} \ne 0$

and the sequences:

$\sequence {x_{k + n} \ {y_{k + n} }^{-1} }$ and $\sequence { {y_{k + n} }^{-1} \ x_{k + n} }$ are well-defined and convergent with:
$\ds \lim_{n \mathop \to \infty} x_{k + n} \ {y_{k + n} }^{-1} = l m^{-1}$
$\ds \lim_{n \mathop \to \infty} {y_{k + n} }^{-1} \ x_{k + n} = m^{-1} l$