Doubts on the proof regarding uniform convergence on complex plane

I want to show $f_n’rightrightarrows f’$ on the region $Omega$.

Let $Omega_delta = {zinOmega:overline{D_delta}(z)subset Omega}$. The textbook I’m reading states that it suffices to show $f_n’rightrightarrows f’$ on $Omega_delta$ for all $delta$. It looks fine at first sight, but I don’t know how it actually makes sense. Can somebody help?