Combination Theorem for Sequences/Complex/Product Rule/Proof 2

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Theorem

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


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:

\(\displaystyle \lim_{n \mathop \to \infty} z_n\) \(=\) \(\displaystyle c\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \lim_{n \mathop \to \infty} x_n + i \lim_{n \mathop \to \infty} y_n\) \(=\) \(\displaystyle a + i b\) $\quad$ Definition of Convergent Complex Sequence $\quad$


\(\displaystyle \lim_{n \mathop \to \infty} w_n\) \(=\) \(\displaystyle d\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \lim_{n \mathop \to \infty} u_n + i \lim_{n \mathop \to \infty} v_n\) \(=\) \(\displaystyle e + i f\) $\quad$ Definition of Convergent Complex Sequence $\quad$


Then:

\(\displaystyle \lim_{n \mathop \to \infty} z_n w_n\) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \paren {\paren {x_n u_n - y_n v_n} + i \paren {y_n u_n + x_n v_n} }\) $\quad$ Definition of Complex Multiplication $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \lim_{n \mathop \to \infty} \paren {x_n u_n - y_n v_n} + i \lim_{n \mathop \to \infty} \paren {y_n u_n + x_n v_n}\) $\quad$ Definition of Convergent Complex Sequence $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {\lim_{n \mathop \to \infty} \paren {x_n u_n} - \lim_{n \mathop \to \infty} \paren {y_n v_n} } + i \paren {\lim_{n \mathop \to \infty} \paren {y_n u_n} + \lim_{n \mathop \to \infty} \paren {x_n v_n} }\) $\quad$ Sum Rule for Real Sequences $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {\lim_{n \mathop \to \infty} \paren {x_n} \lim_{n \mathop \to \infty} \paren {u_n} - \lim_{n \mathop \to \infty} \paren {y_n} \lim_{n \mathop \to \infty} \paren {v_n} } + i \paren {\lim_{n \mathop \to \infty} \paren {y_n} \lim_{n \mathop \to \infty} \paren {u_n} + \lim_{n \mathop \to \infty} \paren {x_n} \lim_{n \mathop \to \infty} \paren {v_n} }\) $\quad$ Product Rule for Real Sequences $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {a e - b f} + i \paren {b e + a f}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {a + i b} \paren {e + i f}\) $\quad$ Definition of Complex Multiplication $\quad$
\(\displaystyle \) \(=\) \(\displaystyle c d\) $\quad$ $\quad$

$\blacksquare$


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