Combination Theorem for Sequences/Normed Division Ring/Multiple Rule
< Combination Theorem for Sequences | Normed Division Ring(Redirected from Multiple Rule for Sequences in Normed Division Ring)
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Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:
- $\ds \lim_{n \mathop \to \infty} x_n = l$
Let $\lambda \in R$.
Then:
- $\sequence {\lambda x_n}$ is convergent and $\ds \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$
Proof
Let
- $\sequence {\tilde{x}_n} := \tuple {\lambda, \lambda, \lambda, \ldots}$
and:
- $\sequence {y_n} := \sequence {x_n}$
The claim follows from Product Rule for Sequences in Normed Division Ring, since:
- $\sequence {\lambda x_n} = \sequence {\tilde{x}_n y_n}$
$\blacksquare$
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