Combination Theorem for Sequences/Real

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

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: