Let $\displaystyle f$ be analytic in $\displaystyle \{ z:0<|z-\alpha |<r\}$ and has a singularity at $\displaystyle \alpha$. Supppose there exists a neighbourhood $\displaystyle \{ z:0<|z-\alpha |<\epsilon \}$ where $\displaystyle Re(f(z))>0$. Show that $\displaystyle \alpha$ is a removable singularity.

I've tried looking at $\displaystyle e^{f(z)}$ in order to show that $\displaystyle f$ is bounded at a neighbourhood of $\displaystyle \alpha$, but all I get is $\displaystyle |e^{f(z)}|=e^{Re(f(z))}>1$ so that dosent help, I also looked at $\displaystyle e^{-f(z)}$ but got nowhere. Need some direction here please.