Let denote the inclusion and denote the projection onto the -axis, so is the continuous projection of onto the -axis. Suppose is a continuous path from to . It follows by the Intermediate value theorem that the image . Let . Clearly , for if it were we could find a such that by the intermediate value theorem. Again invoking the intermediate value theorem, there exists an increasing sequence such that . Now (why?), so by sequential continuity . But is not convergent, since its -coördinate alternates between and , a contradiction. Therefore cannot be a continuous path.