This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
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.