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

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:

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:
 * $\ds \lim_{n \mathop \to \infty} \paren {\lambda z_n + \mu w_n} = \lambda c + \mu d$, $\phi_n \to 0$ 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).