A two-player discrete-time pursuit-evasion game is proposed under some playability conditions where each player's course of action is determined by discrete-time speed controls in the real line. In this regard, two game models are provided as a one-target fashion in discrete-time where the pursuer tries to build a proper control set to capture the target who is the evader in the game, whose ultimate goal is not being caught by his opponent. Each player moves one step at a time and the game ends in favour of the pursuer whenever the evader is trapped within the target region of his opponent. Here, winning strategies are established for the pursuer by designing his finite set which enables him to catch his opponent after a finite number of steps. The main motivation of this work is to investigate a close link between the concept of controllability and two-player games in discrete-time.