▁ 2 ). ▁ ▁The ▁comple ten ess ▁property ▁of ▁the ▁re als ▁is ▁the ▁basis ▁on ▁which ▁calculus , ▁and , ▁more ▁generally ▁mathematical ▁analysis ▁are ▁built . ▁In ▁particular , ▁the ▁test ▁that ▁a ▁sequence ▁is ▁a ▁C auch y ▁sequence ▁allows ▁prov ing ▁that ▁a ▁sequence ▁has ▁a ▁limit , ▁without ▁computing ▁it , ▁and ▁even ▁without ▁knowing ▁it . ▁ ▁For ▁example , ▁the ▁standard ▁series ▁of ▁the ▁exponential ▁function ▁ ▁converges ▁to ▁a ▁real ▁number ▁for ▁every ▁x , ▁because ▁the ▁sums ▁ ▁can ▁be ▁made ▁arbitr arily ▁small ▁( in dep end ently ▁of ▁M ) ▁by ▁choosing ▁N ▁sufficiently ▁large . ▁This ▁proves ▁that ▁the ▁sequence ▁is ▁C auch y , ▁and ▁thus ▁converges , ▁showing ▁that ▁ ▁is ▁well ▁defined ▁for ▁every ▁x . ▁ ▁" The ▁complete ▁ordered ▁field " ▁▁ ▁The ▁real ▁numbers ▁are ▁often ▁described ▁as ▁" the ▁complete ▁ordered ▁field ", ▁a ▁phrase ▁that ▁can ▁be ▁interpreted ▁in ▁several ▁ways . ▁ ▁First , ▁an ▁order ▁can ▁be ▁lattice - complete . ▁It ▁is ▁easy ▁to ▁see ▁that ▁no ▁ordered ▁field ▁can ▁be ▁lattice - complete , ▁because ▁it ▁can ▁have ▁no ▁largest ▁element ▁( g iven ▁any ▁element ▁z , ▁ ▁is ▁larger ), ▁so ▁this ▁is ▁not ▁the ▁sense ▁that ▁is ▁meant . ▁ ▁Additionally , ▁an ▁order ▁can ▁be ▁D ed ek ind - complete , ▁as ▁defined ▁in ▁the ▁section ▁A xi oms . ▁The ▁uniqu eness ▁result ▁at ▁the ▁end ▁of ▁that ▁section ▁just ifies ▁using