▁totally ▁ordered ▁on ▁a ▁number ▁line ▁in ▁a ▁way ▁compatible ▁with ▁addition ▁and ▁multiplication . ▁The ▁second ▁says ▁that , ▁if ▁a ▁non - empty ▁set ▁of ▁real ▁numbers ▁has ▁an ▁upper ▁bound , ▁then ▁it ▁has ▁a ▁real ▁least ▁upper ▁bound . ▁The ▁second ▁condition ▁distingu ishes ▁the ▁real ▁numbers ▁from ▁the ▁rational ▁numbers : ▁for ▁example , ▁the ▁set ▁of ▁rational ▁numbers ▁whose ▁square ▁is ▁less ▁than ▁ 2 ▁is ▁a ▁set ▁with ▁an ▁upper ▁bound ▁( e . g . ▁ 1 . 5 ) ▁but ▁no ▁( r ational ) ▁least ▁upper ▁bound : ▁hence ▁the ▁rational ▁numbers ▁do ▁not ▁satisfy ▁the ▁least ▁upper ▁bound ▁property . ▁ ▁Comple ten ess ▁▁ ▁A ▁main ▁reason ▁for ▁using ▁real ▁numbers ▁is ▁that ▁the ▁re als ▁contain ▁all ▁limits . ▁More ▁precisely , ▁a ▁sequence ▁of ▁real ▁numbers ▁has ▁a ▁limit , ▁which ▁is ▁a ▁real ▁number , ▁if ▁( and ▁only ▁if ) ▁its ▁elements ▁eventually ▁come ▁and ▁remain ▁arbitr arily ▁close ▁to ▁each ▁other . ▁This ▁is ▁formally ▁defined ▁in ▁the ▁following , ▁and ▁means ▁that ▁the ▁re als ▁are ▁complete ▁( in ▁the ▁sense ▁of ▁metric ▁spaces ▁or ▁uniform ▁spaces , ▁which ▁is ▁a ▁different ▁sense ▁than ▁the ▁D ed ek ind ▁comple ten ess ▁of ▁the ▁order ▁in ▁the ▁previous ▁section ). ▁: ▁ ▁A ▁sequence ▁( x n ) ▁of ▁real ▁numbers ▁is ▁called ▁a ▁C auch y ▁sequence ▁if ▁for ▁any ▁ ▁there ▁exists ▁an ▁integer ▁N ▁( poss ibly ▁depending ▁on ▁ ε )