Combination Theorem for Sequences/Complex/Product Rule/Proof 1
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Theorem
- $\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$
Proof
Because $\sequence {z_n}$ converges, it is bounded by Convergent Sequence is Bounded.
Suppose $\cmod {z_n} \le K$ for $n = 1, 2, 3, \ldots$.
Then:
\(\ds \cmod {z_n w_n - c d}\) | \(=\) | \(\ds \cmod {z_n w_n - z_n d + z_n d - c d}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \cmod {z_n w_n - z_n d} + \cmod {z_n d - c d}\) | Triangle Inequality for Complex Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {z_n} \cmod {w_n - d} + m \cdot \size {z_n - c}\) | Complex Modulus of Product of Complex Numbers | |||||||||||
\(\ds \) | \(\le\) | \(\ds K \cdot \cmod {w_n - d} + \cmod d \cdot \cmod {z_n - c}\) | ||||||||||||
\(\ds \) | \(=:\) | \(\ds \phi_n\) |
But $z_n \to c$ as $n \to \infty$.
So $\cmod {z_n - c} \to 0$ as $n \to \infty$ from Convergent Sequence Minus Limit.
Similarly $\cmod {w_n - d} \to 0$ as $n \to \infty$.
From the Combined Sum Rule for Real Sequences:
- $\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$, $\phi_n \to 0$ as $n \to \infty$
The result follows by the Squeeze Theorem for Complex Sequences.
$\blacksquare$