Combination Theorem for Sequences/Complex/Difference Rule
< Combination Theorem for Sequences | Complex(Redirected from Difference Rule for Complex Sequences)
Jump to navigation
Jump to search
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
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Rules. $\text {(ii)}$