Prove that a holomorphic function injective in an annulus is injective in the whole ball
This is quite non-trivial. Suppose $z_1,z_2\in B(0,1)$ are such that $f(z_1)=f(z_2)=z_0$. Take a circle $\gamma\in A$ such that $z_1,z_2\in Int(\gamma)$. Then observe that $f(\gamma)$ is a Jordan curve (this is where you use the injectivity of $f$ in $A$) and thus by Jordan Curve Theorem, $\text{Ind}(f(\gamma),z_0)\leq1$, ignoring orientation. However by the argument principle, $$\text{Ind}(f(\gamma),z_0)=\dfrac{1}{2\pi i} \int_\gamma\dfrac{f'}{f-z_0}\geq2$$
This gives the required contradiction.