Assume that the consumption set X is nonempty, compact, and convex.
Let

denote the budget correspondence. Choose any (p, m) ˆŠ P such that m > minxˆŠX ˆ‘mi=1 Pixi, and let T be an open set such that X(p, m) ˆ© T ‰  Θ. For n = 1, 2,, let
Bn(p, m) = {(p', m') ˆŠ P : || p - p'|| + |m - m'| denote the sequence of open balls about (p, m) of radius 1/n.
1. Show that there exists ˆŠ T such that ˆ‘ni=1 pii 2. Suppose that X(p, m) is not lhc. Show that this implies that
a. There exists a sequence ((pn, mn)) in P such that
(pn, mn) ˆŠ Bn(p, m) and X(pn, mn) ˆ© T = ˆ…
b. There exists N such that ˆŠ X(pN, mN)
c. ˆ‰ T
3. Conclude that X(p, m) is lhc at (p, m).
4. The budget correspondence is continuous for every p ‰  0 such that m > infxˆŠX ˆ‘mi=1 pixxi.

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