how two prove complex sequence is infinite? I have a problem about Polynomial equation and i get the following problem:
"Let $r=a+bi$ with $|r|\le 1$. Prove that $x_1=r, x_{n+1}=(x_n+2)^2, \forall n\in \mathbb{N}$ have infinite element".
It is obvious when $b=0$. When $b\ne 0$. I have proved that $Im x_1< Im x_2 < Im z_3$. However it's hard to continue.
Sombody help me!
 A: $\textbf{Solution.}\quad$ As a more intuitive notation, I will be calling $z_0 = x+iy$ the initial point. To prove that the complex sequence diverges to infinity, I will show that the absolute value increases without bound.
As you mentioned, it is a good idea to start by eliminating the case of $y = 0$, which will give strict inequalities that make this easier.
Consider $z_1 = (x+2 + iy)^2$. The imaginary part can be arbitrarily small, so we will find a bound for $\operatorname{Re} z_1 = (x+2)^2 - y^2$. Since $z_0$ lies in the closed unit disk at the origin, and $y \neq 0$, we see geometrically that $z_0$ cannot be the leftmost point, i.e. $x > -1$. Algebraically $x^2 \leq 1 - y^2 < 1$ also implies $x$ cannot be $-1$. For our target, this means $\operatorname{Re} z_1 > (-1 + 2)^2 - 1^2 = 0$.
We can let $z_1 = p + iq$ where $p > 0$. Then $|z_2| = (p+2)^2 + q^2 > 2^2$, so let $|z_2| = 4 + \varepsilon$ for some $\varepsilon > 0$.
Now, let us show inductively that the absolute value increases exponentially from here, in a way reminiscent of how we bound the Mandelbrot set. Suppose $|z_n| = 4 + \alpha$ for some $\alpha > 0$. Then we have $|z_{n+1}| \geq (2 + \alpha)^2 = 4 + 4\alpha + \alpha^2$, where we have used the fact that adding 2 can decrease the absolute value by at most 2 (if the previous number lies on the negative real axis). This then implies
\begin{equation*}
|z_{n+1}| - 4 > 4\alpha = 4(|z_n| - 4) > 0.
\end{equation*}
Iterating this inequality $k$ times starting from $n=2$, we obtain
\begin{equation*}
\frac{|z_{2+k}| - 4}{|z_2| - 4} > 4^k,
\end{equation*}
which shows that $|z_n|$ increases at least exponentially. This implies the sequence diverges to infinity. $\quad\square$
$\textbf{Remark.}\quad$ You mentioned that the imaginary parts are increasing; however, this is only true if you start with $\operatorname{Im}z_0 > 0$. This does not affect the solution (indeed we could have assumed this WLOG), but please make sure to state your assumptions clearly.
