Let $ \displaystyle{ z_1, z_2 \in \mathbb{C} }$ where $ z_1, z_2 \neq 0$
Prove that: $\displaystyle |z_1 +z_2| \geq \frac{1}{2} \left( |z_1|+|z_2| \right) \left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| $.
P.S I think that I have to use the inequality $ Re(z_1z_2) \leq |z_1||z_2| $ but I don't know how.