Combination Theorem for Sequences/Normed Division Ring/Combined Sum 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:
- $\ds \lim_{n \mathop \to \infty} x_n = l$
- $\ds \lim_{n \mathop \to \infty} y_n = m$
Let $\lambda, \mu \in R$.
Then:
- $\sequence {\lambda x_n + \mu y_n }$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$
Proof
From the Multiple Rule for Sequences in Normed Division Ring, we have:
- $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
- $\ds \lim_{n \mathop \to \infty} \paren {\mu y_n} = \mu m$
The result now follows directly from the Sum Rule for Sequences in Normed Division Ring:
- $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$
$\blacksquare$