What real numbers are in the Mandelbrot set? The Mandelbrot set is defined over the complex numbers and is quite complicated. It's defined by the complex numbers $c$ that remain bounded under the recursion:
$$ z_{n+1} = z_n^2 + c,$$
where $z_1 = 0$.
If $c$ is real, then above recursion will remain real. So for what values of $c$ does the recursion remain bounded?

If z is a complex number whose distance to origin is bigger than $|c|$ and 2 then z is a point than scape for the iteration of the function $z^2+c$. It's easy to demostrate this, then the recursion remain bounded inside the closed ball of radio 2, but we can find the mandelbrot set inside $[-2,0.7]\times[-1.2,1.2]$.
