Combination Theorem for Sequences/Normed Division Ring/Difference Rule

Combination Theorem for Sequences/Normed Division Ring/Difference Rule

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

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.

Also see

 * Sum Rule for Normed Division Ring Sequences
 * Multiple Rule for Normed Division Ring Sequences