Mas-Colell et al. (1995, p. 46) define continuity of preferences as follows:
The preference relation ≿ on X is continuous if it is preserved under limits. That is, for any sequence of pairs (xn; yn) with xn≿yn for all n, with x limn→∞ yn, and limn→∞yn, we have x≿y.
1. Show that this definition in effect requires that the set f{(x; y: x≿y}be a closed subset of X × X.
2. Is this equivalent to the definition given above?