Imagine any finite starting volume D 1 of the phase space and to follow its path under the dynamics of the system. The volume evolves through a "phase tube" in the phase space, keeping its size constant. Assuming a finite phase space, after some number of steps k 1 the phase tube must intersect itself. This means that at least a finite fraction R 1 of the starting volume is recurring. Now, consider the size of the non-returning portion D 2 of the starting phase volume – that portion that never returns to the starting volume. Using the principle just discussed in the last paragraph, we know that if the non-returning portion is finite, then a finite part R 2 of it must return after k 2 steps. But that would be a contradiction, since in a number k 3 = lcm ( k 1 , k 2 ) of step, both R 1 and R 2 would be returning, against the hypothesis that only R 1 was. Thus, the non-returning portion of the starting volume cannot be the empty set, i.e. all D 1 is recurring after some number of steps.