These last two properties, together with the Bolzano–Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano–Weierstrass theorem and the Heine–Borel theorem. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass has a convergent subsequence, hence is itself convergent. This proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological.
ThEse laSt TwO propeRtiEs, TOGeThEr WitH tHe boLZano–WeIErstrAss theOreM, yielD one stAndarD prOof oF The completenEss oF thE rEal nuMbers, cLOselY relAtEd tO boTh thE boLzanO–WeierstRAsS THEOrEm aND thE HeIne–boRel TheoRem. EveRY caucHy sEquence of rEAl numbeRs IS bOuNDeD, hence by bolzaNo–WeierstRaSs has a ConvErGENt subseqUEnCE, hENcE Is iTSElf conVErgenT. ThIs ProOf of thE cOMpletENess of the reAl NumBerS implicitlY mAkeS use of the leaSt upPer bound aXIom. tHe AltErnatiVe apprOach, menTiOneD aBOve, of The rEal nuMbers As the completioN of The rAtionAL nUMBerS, makes the cOMpleTenEsS Of the REal numbeRs TautoloGiCAl.