= Solution
On the <vector space> of nodal values with zero endpoints define the <discrete Sobolev norm>
$$
\boxed{\|W\|_{1,h}=\left(\|W\|_h^2+\|D_x^-W]\!\|_h^2\right)^{1/2}
=\left(h\sum_{i=1}^{N-1}W_i^2+h\sum_{i=1}^N\left(\frac{W_i-W_{i-1}}h\right)^2\right)^{1/2}.}
$$
It 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 $M_4=\max_{0\leq x\leq1}|u^{(4)}(x)|$, finite by the assumed <continuity>. <Taylor's theorem> with fourth-order remainders gives, for suitable $\xi_{i,+}\in(x_i,x_i+h)$ and $\xi_{i,-}\in(x_i-h,x_i)$,
$$
u(x_i\pm h)=u_i\pm hu'_i+\frac{h^2}2u''_i\pm\frac{h^3}6u'''_i+\frac{h^4}{24}u^{(4)}(\xi_{i,\pm}).
$$
Adding the two expansions cancels the odd <derivatives>, so
$$
\varphi_i=\frac{h^2}{24}\left(u^{(4)}(\xi_{i,+})+u^{(4)}(\xi_{i,-})\right),\qquad
|\varphi_i|\leq\frac{M_4h^2}{12}.
$$
Consequently $\|\varphi\|_h^2\leq h(N-1)M_4^2h^4/144\leq M_4^2h^4/144$. Doubling the bound in (c) gives $2\|D_x^-e]\!\|_h^2+\|e\|_h^2\leq\|\varphi\|_h^2$, and therefore
$$
\boxed{\|U-u\|_{1,h}\leq\|\varphi\|_h\leq\frac{M_4}{12}h^2.}
$$
Thus $M_4/12$ is a valid nonnegative constant. To satisfy the request for a strictly positive constant even when $u^{(4)}\equiv0$, take \b[$C=\max\{1,M_4\}/12$], 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.
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