Combination Theorem for Sequences/Normed Division Ring/Difference 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:

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

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

Then:

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

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

$\ds \lim_{n \mathop \to \infty} \paren {-y_n} = -m$

Hence:

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

The result follows.

$\blacksquare$