A mountain climber wants to climb up a mountain that is 1 000 meters high. In the first hour, he climbed half the height of the mountain. In the second hour, he climbed half the distance he climbed in the first hour. In the third hour, he climbed half the distance he climbed in the second hour and so on. Determine whether the mountain climber will eventually be able to reach the peak of the mountain or not. If so, determine how many hours will he need to reach the peak. If not, you must explain why. Relate this to the concept of limits.