A simple proof is given to show that iterates of the complex quadratic map would eventually escape to infinity if |z_k| > r(c) where r(c) = max(|c|, 2).

Assumptions:
For a given iterate, k,

• |z_k| ³ |c|
• |z_k| > 2

Assertion:
There exists a positive number e such that |z_k+1| = (1 + e)|z_k|

It follows from the above assertion that the m^th iterate of z_k will at least be (1+e)^m times as large as z_k in magnitude, implying that the point is attracted to infinity.

Proof:
From triangle inequality, it follows that

|z2| = |z2 + c - c| £ |z2 + c| + |c| (1)

Solving the above inequality for |z² + c| gives

|zk2 +c| ³ |zk2|- |c| = |zk|2 - |c| ³ |zk|2 - |zk| using assumption 1 = (|zk| - 1)|zk| |zk+1| = (1 + e)|zk| using assumption 2