Combination Theorem for Sequences/Normed Division Ring/Combined Sum 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:


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

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

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

Proof
From the Multiple Rule for Normed Division Ring Sequences, we have:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {\mu y_n} = \mu m$

The result now follows directly from the Sum Rule for Normed Division Ring Sequences:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$

Also see

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