Imagine a standard ruler (made of atoms) at the present epoch. Assume its comoving length is [itex]dx=x_1 - x_2[/itex] where [itex]x_1[/itex] and [itex]x_2[/itex] are the comoving coordinates of its ends at the present time. As the scale factor [itex]a=1[/itex] then its proper length [itex]ds=a \ dx[/itex] is equal to its comoving length [itex]dx[/itex]. Now imagine that ruler persists to a later epoch with [itex]a=2[/itex]. As the ruler doesn't expand with the Universe is it correct to say that its proper length [itex]ds[/itex] is still equal to its comoving length [itex]dx[/itex] even though the space around it has expanded by a factor of two?