Uniform utility convergence need not preserve closed preference convergence

On $[0,1]$ the utility functions $u_n(x)=x/n$ converge uniformly on the whole domain to $u(x)=0$. Their preference relations remain the usual order, while the limiting utility function makes all pairs indifferent. The weak comparison $0⪰1$ in the limit cannot be approximated by comparisons from the varying graphs. Uniform convergence does guarantee the upper graph-limit condition, because convergent pairs remain in a fixed bounded set.