Real Slice of Mandelbrot set

I'm trying to prove that the real slice of a Mandelbrot set is M n R =[-2,1/4]

I've tried researching it on the internet but everything I find seems to just state this and not prove it.

The hint I've been given to start is is to let

$\displaystyle c$\in$$(1/4, $\displaystyle $\infty$$) proving that for each $\displaystyle $z$$\in$R$ that $\displaystyle f$_{\text{c}}$(z)-z $\geq$ c -$1/4

where $\displaystyle f$_{\text{c}}$(z)= z^2+c$

Any help would be greatly appreciated

Re: Real Slice of Mandelbrot set

What you have written makes no sense. For any real number c, f_{c}(z)- z is a complex number. You cannot say it is "larger than c- 1/4, a real number.

Perhaps you meant $\displaystyle \left|f_c(z)- z\right|\ge c- \frac{1}{4}$?