What are the bounds of the Mandelbrot set? When does it diverge?
A6d: The Mandelbrot set lies within |c| <= 2. If |z| exceeds 2, the z sequence diverges. Proof: if |z| > 2, then |z^2 + c| >= |z^2| – |c| > 2|z| – |c|. If |z| >= |c|, then 2|z| – |c| > |z|. So, if |z| > 2 and |z| >= c, then |z^2 + c| > |z|, so the sequence is increasing. (It takes a bit more work to prove it is unbounded and diverges.) Also, note that |z| = c, so if |c| > 2, the sequence diverges.
