Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 36 / 5 / b

There are two equivalent graph-limit requirements. Both are necessary. The upper condition is that, for every subsequence $n_j$ and convergent pairs $x_j\to x$, $y_j\to y$,The lower condition is that every limiting comparison can be approximated: for every $x⪰y$, there exist $x_n\to x$ and $y_n\to y$ withThe upper condition excludes spurious limiting comparisons; the lower condition prevents genuine limiting comparisons from disappearing. The subsequence quantifier in the upper condition is important: a violation need occur along only some of the indices. On the locally compact separable metric space $X\times X$, these two conditions are exactly closed convergence of preference relations.