If $|z| < 1$, prove that $\Re \left(\frac{1}{1 - z} \right) > \frac{1}{2}$. 
If $|z| < 1$, prove that $\Re \left(\frac{1}{1 - z} \right) > \frac{1}{2}$.

My attempt:
Consider $\frac{1}{1 - z}$. Let $z = x + iy$, we know that $|z| < 1 \implies x, y < 1$. $$\frac{1}{1 - z} = \frac{1}{1 - x - iy} = \frac{1 - x + iy}{(1 - x)^2 + y^2}.$$
$$\Re \left(\frac{1}{1 - z} \right) = \frac{1 - x}{(1 - x)^2 + y^2}.$$
I got $$\frac{1}{(x-1)^2 + y^2} > \frac{1}{2}.$$
How do I manage the numerator? Can you help me? I welcome the alternative approaches.
 A: You need to use more than just $x,y<1$ for example with $x=y=\frac{3}{4}$ you obtain $\Re\left(\frac{1}{1-(x+iy)} \right)=\frac{2}{5} <\frac{1}{2}$.
Hint:
Using your computations:
$$ \Re\left(\frac{1}{1-z}\right)=\frac{1-x}{(1-x)^2+y^2}=\frac{1-x}{x^2+y^2-2x+1}$$
but $x^2+y^2<1$ so:
$$\frac{1-x}{x^2+y^2-2x+1}>\frac{1-x}{1-2x+1}=\frac{1}{2}$$
A: $$\Re \Big(\frac{1}{1-z}\Big)={1\over 2}\Big(\frac{1}{1-z}+\frac{1}{1-\overline{z}}\Big)$$
$$={1\over 2}\frac{1-\overline{z}+1-z}{1-z-\overline{z}+z\overline{z}}$$
$$>{1\over 2}\frac{1-\overline{z}+1-z}{1-z-\overline{z}+1}$$
$$={1\over 2}$$
A: I have also a pure geometric solution. 
All points $z$ satisfying $|z|<1$ are ''inside'' of circle $\mathcal{C}$ with radius $r=1$ and center at $0$. 
Now transformation $z\mapsto 1-z$ is reflection across $0$ followed by translation for $1$. So $\mathcal{C}$ ''moves'' to the right for $1$ and let this new circle be $\mathcal{C}'$. 
Transformation $z\mapsto {1\over z}$ is inversion with center at $0$ with radius $1$ (i.e. circle $\mathcal{C}$). Since $0\in \mathcal{C}'$ and $\mathcal{C}'$ touches imaginary axis, $\mathcal{C}'$ maps to line $\ell$ which is parallel to imaginary axis and all the points ''inside'' of circle $\mathcal{C}'$ goes ''right'' to the $\ell$. Finally, since points of intersection of $\mathcal{C}$ and $\mathcal{C}'$, which have clearly real part ${1\over 2}$, are invariant, the line $\ell$ has equation $z={1\over 2}$ and thus conclusion.
