Combination Theorem for Sequences/Complex/Product Rule/Proof 2
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Theorem
- $\ds \lim_{n \mathop \to \infty} \paren {z_n w_n} = c d$
Proof
Let $z_n = x_n + i y_n$.
Let $w_n = u_n + i v_n$.
Let $c = a + i b$
Let $d = e + i f$.
By definition of convergent complex sequence:
\(\ds \lim_{n \mathop \to \infty} z_n\) | \(=\) | \(\ds c\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{n \mathop \to \infty} x_n + i \lim_{n \mathop \to \infty} y_n\) | \(=\) | \(\ds a + i b\) | Definition of Convergent Complex Sequence |
\(\ds \lim_{n \mathop \to \infty} w_n\) | \(=\) | \(\ds d\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{n \mathop \to \infty} u_n + i \lim_{n \mathop \to \infty} v_n\) | \(=\) | \(\ds e + i f\) | Definition of Convergent Complex Sequence |
Then:
\(\ds \lim_{n \mathop \to \infty} z_n w_n\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \paren {\paren {x_n u_n - y_n v_n} + i \paren {y_n u_n + x_n v_n} }\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \paren {x_n u_n - y_n v_n} + i \lim_{n \mathop \to \infty} \paren {y_n u_n + x_n v_n}\) | Definition of Convergent Complex Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\lim_{n \mathop \to \infty} \paren {x_n u_n} - \lim_{n \mathop \to \infty} \paren {y_n v_n} } + i \paren {\lim_{n \mathop \to \infty} \paren {y_n u_n} + \lim_{n \mathop \to \infty} \paren {x_n v_n} }\) | Sum Rule for Real Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\lim_{n \mathop \to \infty} \paren {x_n} \lim_{n \mathop \to \infty} \paren {u_n} - \lim_{n \mathop \to \infty} \paren {y_n} \lim_{n \mathop \to \infty} \paren {v_n} } + i \paren {\lim_{n \mathop \to \infty} \paren {y_n} \lim_{n \mathop \to \infty} \paren {u_n} + \lim_{n \mathop \to \infty} \paren {x_n} \lim_{n \mathop \to \infty} \paren {v_n} }\) | Product Rule for Real Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a e - b f} + i \paren {b e + a f}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a + i b} \paren {e + i f}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds c d\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Rules