# functional analysis – Norm of a linear form in a function space

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?