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:

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


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:

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

$\blacksquare$

Also see

Sources