If there is not a smallest actual unit of space, then it follows that there are an infinite amount of subdivisions of space actually existing in any given finite region of space. If this is so, then it also follows that there is an infinite amount of space in any finite region of space. Now, it would seem that it would take an infinite amount of time to move through an infinite amount of space, unless one has the capacity to move at an infinite velocity. Since human beings do not seem to possess the means to means to travel at an infinite velocity, it follows that no one would be able to cross even a finite distance, since to do so they would have to move through the infinite subdivisions of the finite amount of space. In fact, nothing unable to move at an infinite velocity would be able to move at all, since it would have to cross infinite space to reach a point even infinitesimally farther away from where it started.
However, since clearly things do move, it follows that space cannot be actually infinitely subdivided. If space is not actually infinitely subdivided, it must be finitely subdivided; therefore, it is necessary that there be a smallest unit of space. If there is a smallest unit of space, then it follows that there must also be a smallest possible unit of matter or energy, though it is entirely possible that this unit may be too small to be detected.