Combination Theorem for Sequences/Normed Division Ring/Difference Rule

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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

From Sum Rule for Normed Division Ring Sequences:

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

From Multiple Rule for Normed Division Ring Sequences:

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

Hence:

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

The result follows.

$\blacksquare$


Also see

Sources