Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percentage (other than 0% and 100%) from the middle. In the case where the middle of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0s and 9s. If a fixed percentage is removed at each stage, then the limiting set will have measure zero, since the length of the remainder as for any such that .
instEaD of rEpEatedLy rEMoVing THE Middle ThiRd of eVery PIece As in The CanTOr set, we coulD aLso KeEp remOving any other fiXEd peRcEntage (otheR thAn 0% and 100%) from the midDle. in the casE whErE the mIddlE oF The iNterVal is remoVed, we GeT a REmarKaBLy ACcessiBle cAse — THE Set cOnsists oF all NuMberS in [0,1] that can BE written As A deCiMal consISting ENtirely Of 0s aNd 9S. if a fixed percenTAge Is rEmOvED at each StagE, then ThE lImiTinG set Will havE mEAsure zero, sInce The LeNGTh of the reMainDEr as fOr Any SUCh THAT .