Two methods are proposed for successive approximations of the value function of a pursuit game with limited time and with payoff function min_{τε{lunate}[0, t]} H (x (τ), y (τ)), which are used directly for the constructive design of successive pursuit and evasion strategies which permit ε-optimal strategies to be found in any ε > c0. Necessary and sufficient conditions for some function to be the value function of the pursuit game being analyzed are derived as well. References [1-3] were devoted to sequential methods of constructing the value function or the payoff minimax in game problems of encounter at a specified instant. These methods were used in [1, 2] to construct maximal stable bridges the strategies extremal to which solve the corresponding problem, as is well known from [4], Sequential procedures for constructing maximal stable bridges without a preliminary construction of the value function or the payoff minimax also were examined in [1,2] and in [5].