# 6d: What are the bounds of the Mandelbrot set? When does it diverge?

## Description

This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by
others.

# 6d: What are the bounds of the Mandelbrot set? When does it diverge?

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, |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 |z1=c, so if |c|>2, the sequence diverges.

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