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

## 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 $\quad$ $\displaystyle$ $=$ $\displaystyle c d$ $\quad$ $\quad$

$\blacksquare$