Proof that $\bar D(-1,\frac14)$ lies inside the Mandelbrot set

Question

I am currently compiling proofs of the most elementary properties of the Mandelbrot set $M$.

For example :

• $M\cap\mathbb{R}=[-2,\frac14]$
• $\bar D(0;\frac14)\subset M$.
• $M\subset\bar D(0,2)$
• $c\in M\implies K_c\subset\bar D(0,2)$ (where $K_c$ denotes the filled Julia set associated to $c$)

Those proofs are very simple and involve essentially the triangle inequality.

I would like to know if I can expect an elementary proof for : $\bar D(-1,\frac14)\subset M$.