▁such ▁that ▁the ▁distance ▁ ▁is ▁less ▁than ▁ ε ▁for ▁all ▁n ▁and ▁m ▁that ▁are ▁both ▁greater ▁than ▁N . ▁This ▁definition , ▁originally ▁provided ▁by ▁C auch y , ▁formal izes ▁the ▁fact ▁that ▁the ▁x n ▁eventually ▁come ▁and ▁remain ▁arbitr arily ▁close ▁to ▁each ▁other . ▁ ▁A ▁sequence ▁( x n ) ▁converges ▁to ▁the ▁limit ▁x ▁if ▁its ▁elements ▁eventually ▁come ▁and ▁remain ▁arbitr arily ▁close ▁to ▁x , ▁that ▁is , ▁if ▁for ▁any ▁ ▁there ▁exists ▁an ▁integer ▁N ▁( poss ibly ▁depending ▁on ▁ ε ) ▁such ▁that ▁the ▁distance ▁ ▁is ▁less ▁than ▁ ε ▁for ▁n ▁greater ▁than ▁N . ▁ ▁Every ▁conver gent ▁sequence ▁is ▁a ▁C auch y ▁sequence , ▁and ▁the ▁con verse ▁is ▁true ▁for ▁real ▁numbers , ▁and ▁this ▁means ▁that ▁the ▁topological ▁space ▁of ▁the ▁real ▁numbers ▁is ▁complete . ▁ ▁The ▁set ▁of ▁rational ▁numbers ▁is ▁not ▁complete . ▁For ▁example , ▁the ▁sequence ▁( 1 ; ▁ 1 . 4 ; ▁ 1 . 4 1 ; ▁ 1 . 4 1 4 ; ▁ 1 . 4 1 4 2 ; ▁ 1 . 4 1 4 2 1 ; ▁... ), ▁where ▁each ▁term ▁adds ▁a ▁digit ▁of ▁the ▁decimal ▁expansion ▁of ▁the ▁positive ▁square ▁root ▁of ▁ 2 , ▁is ▁C auch y ▁but ▁it ▁does ▁not ▁conver ge ▁to ▁a ▁rational ▁number ▁( in ▁the ▁real ▁numbers , ▁in ▁contrast , ▁it ▁converges ▁to ▁the ▁positive ▁square ▁root ▁of