Combination Theorem for Sequences/Complex/Product Rule

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Theorem

Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.

Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:

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


Then:

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


Proof 1

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).


$\blacksquare$


Proof 2

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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