Combination Theorem for Sequences/Complex/Difference Rule

Theorem
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:


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

Then:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {z_n - w_n} = c - d$

Proof
From Sum Rule for Complex Sequences:


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

From Multiple Rule for Sequences:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {-w_n} = -d$

Hence:
 * $\displaystyle \lim_{n \mathop \to \infty} \paren {z_n + \paren {-w_n} } = c + \paren {-d}$

The result follows.

Also see

 * Sum Rule for Complex Sequences