Combination Theorem for Sequences/Sum Rule

Theorem

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:

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

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

Then:

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

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:

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

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

Then:

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