Why c>1/4 is not in Mandelbrot set

Question

As title:

$f_c(x)=x^2+c$

I got to the step:

$f_c(x)>x$ (for all x)

But what's next? How to show that after k iterations, $f^k_c \to \infty$ as $k \to \infty$

Thanks,

Answer 1

Let $c={1\over4}+\epsilon$, and note that

$$f_c(x)=x^2+c=x+\left(x-{1\over2}\right)^2+\epsilon\ge x+\epsilon$$

If $\epsilon\gt0$, it follows that

$$f_c^{(k)}(x)\ge x+k\epsilon\to\infty$$

as $k\to\infty$.