Combination Theorem for Sequences/Combined Sum Rule

Combined Sum Rule for Real Sequences
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:


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

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

Then:

Combined Sum Rule for Complex Sequences
Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.

Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:


 * $\ds \lim_{n \mathop \to \infty} z_n = c$
 * $\ds \lim_{n \mathop \to \infty} w_n = d$

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

Then: