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:

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

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

Then:

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

Proof

From Sum Rule for Complex Sequences:

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

From Multiple Rule for Complex Sequences:

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

Hence:

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

The result follows.

$\blacksquare$

Also see

• Sum Rule for Complex Sequences

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Rules. $\text {(ii)}$