We consider infinite, two person zero sum games played as follows: On the nth move, players A, B select privately from fixed finite sets, A,, Bn, the result of their selections being made known before the next selection is made. A point in the associated sequence space Q = ]7n= (An x B,n) is thus produced upon which B pays A an amount determined by a payoff function defined on U. We show that if the payoff functions of games {GnJ are upper semicontinuous and decrease pointwise to a function which is the payoff for a game, G, then Val(GJ)IVal(G). This shows that a certain class of games can be approximated by finite games. We then give a counterexample to possibly more general approximation theorems by displaying a sequence of games {GJ} with upper semicontinuous payoff functions which increase to the payoff of a game G, and where Val(Gn)=0 for all n but Val(G) =