# Combination Theorem for Sequences/Complex/Product Rule/Proof 1

## Theorem

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

## 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:

\(\displaystyle \cmod {z_n w_n - c d}\) | \(=\) | \(\displaystyle \cmod {z_n w_n - z_n d + z_n d - c d}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \cmod {z_n w_n - z_n d} + \cmod {z_n d - c d}\) | $\quad$ Triangle Inequality for Complex Numbers | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \cmod {z_n} \cmod {w_n - d} + m \cdot \size {z_n - c}\) | $\quad$ Complex Modulus of Product of Complex Numbers | $\quad$ | |||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle K \cdot \cmod {w_n - d} + \cmod d \cdot \cmod {z_n - c}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=:\) | \(\displaystyle \phi_n\) | $\quad$ | $\quad$ |

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:

- $\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$, $\phi_n \to 0$ as $n \to \infty$

The result follows by the Squeeze Theorem for Sequences of Complex Numbers (which applies as well to real as to complex sequences).

$\blacksquare$