▁with ▁an ▁upper ▁bound ▁in ▁R ▁has ▁a ▁least ▁upper ▁bound ▁( also ▁called ▁supre m um ) ▁in ▁R ▁. ▁ ▁The ▁last ▁property ▁is ▁what ▁differenti ates ▁the ▁re als ▁from ▁the ▁rational s ▁( and ▁from ▁other , ▁more ▁ex otic ▁ordered ▁fields ). ▁For ▁example , ▁the ▁set ▁of ▁rational s ▁with ▁square ▁less ▁than ▁ 2 ▁has ▁a ▁rational ▁upper ▁bound ▁( e . g ., ▁ 1 . 5 ) ▁but ▁no ▁rational ▁least ▁upper ▁bound , ▁because ▁the ▁square ▁root ▁of ▁ 2 ▁is ▁not ▁rational . ▁ ▁These ▁properties ▁imply ▁Arch im ede an ▁property ▁( which ▁is ▁not ▁imp lied ▁by ▁other ▁definitions ▁of ▁comple ten ess ). ▁That ▁is , ▁the ▁set ▁of ▁integers ▁is ▁not ▁upper - bounded ▁in ▁the ▁re als . ▁In ▁fact , ▁if ▁this ▁were ▁false , ▁then ▁the ▁integers ▁would ▁have ▁a ▁least ▁upper ▁bound ▁N ; ▁then , ▁N ▁– ▁ 1 ▁would ▁not ▁be ▁an ▁upper ▁bound , ▁and ▁there ▁would ▁be ▁an ▁integer ▁n ▁such ▁that ▁, ▁and ▁thus ▁, ▁which ▁is ▁a ▁contradiction ▁with ▁the ▁upper - bound ▁property ▁of ▁N . ▁ ▁The ▁real ▁numbers ▁are ▁uniqu ely ▁specified ▁by ▁the ▁above ▁properties . ▁More ▁precisely , ▁given ▁any ▁two ▁D ed ek ind - complete ▁ordered ▁fields ▁R 1 ▁and ▁R 2 , ▁there ▁exists ▁a ▁unique ▁field ▁isomorphism ▁from ▁R 1 ▁to ▁R 2 , ▁allowing ▁us ▁to ▁think ▁of ▁them ▁as ▁essentially ▁the ▁same ▁mathematical ▁object . ▁ ▁For