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 $\ds$ $=$ $\ds c d$

$\blacksquare$