It's easy to think of infinite counting upwards as having to come to an end eventually, but what about dividing a thing down? Can we have smaller and smaller fr

It's easy to think of infinite counting upwards as having to come to an end eventually, but what about dividing a thing down? Can we have smaller and smaller fractions until we have an infinity of tiny bits, or does it go on for so long there is a point when all of the bits are of size 0? Anyways, I enjoyed reading the perspective of a mathematician on this.