We can prove the existence of atoms, the smallest unit of matter, by mere logic, via reductio ad absurdum.

Assume that matter can be divided infinitely. Then each of the infinitely many resulting parts must either have no magnitude or some magnitude. If the parts have no magnitude, then even an infinite number of them could never compose anything with a finite size: zero added infinitely many times is still zero. If the parts have some magnitude, then an infinite number of nonzero parts would yield an infinite total size, which contradicts the observable fact that matter occupies a finite quantity of space. Therefore, the assumption of infinite divisibility must be false.