The paper gives sufficient conditions for the existence of Nash equilibrium points in a class of n-person noncooperative games with the pay-offs which are the limits of decreasing sequences of continuous functions. A situation   in the game can be characterized by a set of functions each being defined on the set of situations  . Putting   the product element   may be associated with each situation. The set   with a topology of simple convergence is said to be,N continuous at the point  , if there exists a sequence   for any ε> 0; which converges to ξ in   that is   where D_ε (ξ_n) is a set of points in which the variation of the function ξ_n is not less than ε. The following theorem is proved: Let X_i be compact spaces, H_n be limited BAIRE upper semi-continuous functions spaces,   be N-continuous in each point. Then the game Г possesses an equilibrium point in mixed strategies. - The proof of the theorem is based on a. number of auxiliary propositions; given is the paper and is illustrated by an example.