Combination Theorem for Sequences/Real/Combined Sum Rule

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Theorem

Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent 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:

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


Proof

From the Multiple Rule for Real 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 Real Sequences:

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

$\blacksquare$


Sources