Properties of Convergent Sequences of Complex Numbers

# Properties of Convergent Sequences of Complex Numbers

We will now state some properties of convergent sequences of complex numbers. All of these statements are proved analogously to the proofs for the real cases.

 Theorem 1: If $(z_n)_{n=1}^{\infty}$ is a sequence of complex numbers that converges then its limit is unique.
• Proof: Suppose that $(z_n)_{n=1}^{\infty}$ converges to both $A$ and $B$. Let $\epsilon > 0$ be given.
• Then by definition, for $\epsilon_1 = \frac{\epsilon}{2} > 0$ there exists an $N_1 \in \mathbb{N}$ such that if $n \geq N_1$ then:
(1)
\begin{align} \quad |z_n - A| < \epsilon_1 = \frac{\epsilon}{2} \quad (*) \end{align}
• Similarly, for $\epsilon_2 = \frac{\epsilon}{2} > 0$ there exists an $N_2 \in \mathbb{N}$ such that if $n \geq N_2$ then:
(2)
\begin{align} \quad |z_n - B| < \epsilon_2 = \frac{\epsilon}{2} \quad (**) \end{align}
• Let $N = \max \{ N_1, N_2 \}$. Then if $n \geq N$, both $(*)$ and $(**)$ hold and so:
(3)
\begin{align} \quad |A - B| = |A - z_n + z_n - B| \leq |A - z_n| + |z_n - B| < \epsilon_1 + \epsilon_2 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}
• Since $|A - B| < \epsilon$ for all $\epsilon > 0$ be must have that $|A - B| = 0$. So $A - B = 0$, i.e., $A = B$. So if $(z_n)_{n=1}^{\infty}$ converges, it's limit is unique. $\blacksquare$
 Theorem 2: Let $(z_n)_{n=1}^{\infty}$ and $(w_n)_{n=1}^{\infty}$ be sequences of real numbers that converge to $Z$ and $W$ respectively. Then: a) $(z_n + w_n)_{n=1}^{\infty}$ converges to $Z + W$. b) $(z_n - w_n)_{n=1}^{\infty}$ converges to $Z - W$. c) $(z_nw_n)_{n=1}^{\infty}$ converges to $Z \cdot W$. d) $\displaystyle{\left ( \frac{z_n}{w_n} \right )_{n=1}^{\infty}}$ converges to $\displaystyle{\frac{Z}{W}}$ provided that $W \neq 0$.
• Proof of a) Let $\epsilon > 0$ be given. By definition, since $(z_n)_{n=1}^{\infty}$ converges to $Z$, for $\epsilon_1 = \frac{\epsilon}{2} > 0$ there exists an $N_1 \in \mathbb{N}$ such that if $n \geq N_1$ then:
(4)
\begin{align} \quad |z_n - Z| < \epsilon_1 = \frac{\epsilon}{2} \quad (*) \end{align}
• Similarly, since $(w_n)_{n=1}^{\infty}$ converges to $W$, for $\epsilon_2 = \frac{\epsilon}{2} > 0$ there exists an $N_2 \in \mathbb{N}$ such that if $n \geq N_2$ then:
(5)
\begin{align} \quad |w_n - W| < \epsilon_2 = \frac{\epsilon}{2} \quad (**) \end{align}
• Let $N = \max \{ N_1, N_2 \}$. Then if $n \geq N$, both $(*)$ and $(**)$ hold and:
(6)
\begin{align} \quad |z_n + w_n - (Z + W)| = |z_n - Z + w_n - W| \leq |z_n - Z| + |w_n - W| < \epsilon_1 + \epsilon_2 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}
• Therefore $(z_n + w_n)_{n=1}^{\infty}$ converges to $Z + W$. $\blacksquare$