 Solution

On the vector space of nodal values with zero endpoints define the discrete Sobolev normIt is a norm because it is the Euclidean norm of the scaled list of nodal values and first finite differences, and vanishes only for the zero list. It is the discrete counterpart of the Sobolev space $H^1$ norm. Here $W=U-u$ has zero endpoints, so this definition applies directly.

Set $M_4={\max }_{0\le x\le 1}|u^{(4)}(x)|$, finite by the assumed continuity. Taylor's theorem with fourth-order remainders gives, for suitable ${\xi }_{i,+}\in (x_i,x_i+h)$ and ${\xi }_{i,-}\in (x_i-h,x_i)$,Adding the two expansions cancels the odd derivatives, soConsequently $\Vert \phi {\Vert }_h^2\le h(N-1)M_4^2{h}^4/144\le M_4^2{h}^4/144$. Doubling the bound in (c) gives $2\Vert D_x^{-}e]{\Vert }_h^2+\Vert e{\Vert }_h^2\le \Vert \phi {\Vert }_h^2$, and thereforeThus $M_4/12$ is a valid nonnegative constant. To satisfy the request for a strictly positive constant even when $u^{(4)}\equiv 0$, take $C=\max \{1,M_4\}/12$, independent of $h$. This proves second-order convergence of a numerical method in a discrete Sobolev norm, including control of the backward difference operator applied to the error.