I would appreciate some help with the following problem. Let $u, u_n:C((0,1))tomathbb{R}$ the linear forms defined by $$u(f)=int_0^1f(x)dxquad&quad u_n(f)=frac{1}{n}sum_{k=1}^nfleft(frac{k}{n}right).$$ I’m trying to prove that $$||u_n-u||=sup_{||f||_infty=1}|u_n(f)-u(f)|=2.$$

I already have that $||u_n-u||leq2$ and I’ve tried to show that the $sup$ is attained (I’m not sure that it is true) by using trigonometric functions but I haven’t been able to get anything. Could anyone give me a hint?

Thanks in advance.