Combination Theorem for Sequences

Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be sequences in $X$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be convergent to the following limits:

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

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

Let $\lambda, \mu \in X$.

Then the following results hold:

Sum Rule

$\displaystyle\lim_{n \to \infty} \left({x_n + y_n}\right) = l + m$

Multiple Rule

$\displaystyle\lim_{n \to \infty} \left({\lambda x_n}\right) = \lambda l$

Combined Sum Rule

$\displaystyle\lim_{n \to \infty} \left({\lambda x_n + \mu y_n}\right) = \lambda l + \mu m$

Product Rule

$\displaystyle\lim_{n \to \infty} \left({x_n y_n}\right) = l m$

Quotient Rule

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

provided that $m \ne 0$.