Rangarira ndima iri pasi apa uye ipapo pindura mubvunzo gare gare mumutauro mumwe chetewo sechikamu:  Kupfuura: Muchimiro ichi, zvizivikanwe 1 = 0.999... uye 1 = 1.000... zvinoratidzira, zvichiteerana, chokwadi chokuti 1 iri zvose zviri zviviri [0, 1] uye [1, 2], naizvozvo munhu anogona kusarudza chero chikamu chipi nechipi paanenge achiwana manhamba ayo. Kuti ave nechokwadi chokuti chiratidzo ichi hachishandisi zvisina kufanira chiratidzo che "=" munhu anoda nzira yokugadzirazve nhamba chaiyo chaiyo nokuda kwechikamu chimwe nechimwe chedesimal. Izvi zvinogona kuitwa nemiganhu, asi zvimwe zvivakwa zvinopfuurira nenyaya yekuronga. Imwe sarudzo yakananga ndeye nested intervals theorem, iyo inovimbisa kuti kupihwa kuteedzana kwe nested, yakavharwa nguva iyo urefu hunova hudiki, nguva yacho ine imwe chaiyo chaiyo mukuyambuka kwavo. Saka b0.b1b2b3... inotsanangurwa kuva nhamba yakasarudzika iri mukati mezvikamu zvose [b0, b0 + 1], [b0.b1, b0.b1 + 0.1], zvichingodaro. 0.999... ipapo inhamba chaiyo yakasarudzika iri mukati mezvikamu zvose [0, 1], [0.9, 1], [0.99, 1], uye [0.99...9, 1] nokuda kweshoko rimwe nerimwe rakaperera re9s. Sezvo 1 chiri chinhu chimwe nechimwe chezvikamu izvi, 0.999... = 1. The Nested Intervals Theorem inowanzosimbiswa pane chimwe chinhu chinokosha chezviverengero chaizvoizvo: kuvapo kwemiganhu yapamusoro kana kuti yakakwirira. Kuti ashandise zvakananga zvinhu izvi, munhu angatsanangura b0.b1b2b3... kuva muganhu wokumusoro muduku wechikwata chevanenge vari pedyo {b0, b0.b1, b0.b1b2,...}. Mumwe anogona ipapo kuratidza kuti iyi rondedzero (kana rondedzero yematanho akaiswa) inowirirana nenzira yekuparadzanisa, zvichireva 0.999... = 1 zvakare. Tom Apostol anogumisa,  Mubvunzo: Ndechipi chiri chimiro chechiverengero chakadudzwa?
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