Just note that $\left| {\left[ {f - g} \right](z) - \left[ {f - g} \right](z')} \right| \leqslant \left| {f(z) - f(z')} \right| + \left| {g(z) - g(z')} \right|$.
Each of the last two can be made $< \frac{\epsilon }{2}$.