Combination Theorem for Sequences/Real

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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 the following results hold:


Sum Rule

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


Difference Rule

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


Multiple Rule

$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$


Combined Sum Rule

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


Product Rule

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


Quotient Rule

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

provided that $m \ne 0$.