= Uniform utility convergence need not preserve closed preference convergence
On $[0,\infty)$ the <utility functions> $u_n(x)=x/n$ converge uniformly on every bounded set to $u(x)=0$. Their <preference relations> remain the usual order, while the limiting <utility function> makes all pairs indifferent. The weak comparison $0\succeq1$ 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.
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