Say that we have a line from 0 to 0.5. I want a process that can split that line and half and move that half a bit, and then take half of that half and moved it and so on, so that by the end all the pieces of the line would still be within the region of 0 to 1.

An example of such a series is to take half of the line and move it half its size: (0, 0.5) (0, 0.25), (0.375, 0.625) (0, 0.25), (0.375, 0.5), (0.5625, 0.6875)

1. Is there a name for such a series? (it is similar to a cantor set, but not quite)
2. How does one prove that this set will never reach 1, no matter how many splits we'll make?