Hint:
Prove: $\displaystyle (f_n)\rightarrow f \Rightarrow (\Delta f_n)\rightarrow \Delta f$.
Regards.
Fernando Revilla
Personally while the other two responses are correct I think it's easier to notice that if $\displaystyle \|f-g\|_{\infty}<\frac{\varepsilon}{4}$ then $\displaystyle \displaystyle \left|f(1)-g(1)+(f(0)-g(0))-(2f(\frac{1}{2})-g(\frac{1}{2}))\right|\leqslant |f(1)-g(1)|+|f(0)-g(0)|+2|f(\frac{1}{2})-g(\frac{1}{2})|\leqslant 4\|f-g\|_{\infty}<\varepsilon$.