▁has ▁no ▁repet ition , ▁it ▁is ▁possible ▁to ▁estimate ▁each ▁q n ▁against ▁the ▁series ▁ ▁Thus ▁the ▁sequence ▁( q n ) ▁is ▁bounded ▁above ▁by ▁ 1 . ▁ ▁Class ically , ▁this ▁means ▁that ▁q n ▁has ▁a ▁supre m um ▁x . ▁▁ ▁It ▁is ▁shown ▁that ▁x ▁is ▁not ▁a ▁comput able ▁real ▁number . ▁The ▁proof ▁uses ▁a ▁particular ▁fact ▁about ▁comput able ▁real ▁numbers . ▁If ▁x ▁were ▁comput able ▁then ▁there ▁would ▁be ▁a ▁comput able ▁function ▁r ( n ) ▁such ▁that ▁| q j ▁- ▁q i | ▁< ▁ 1 / n ▁for ▁all ▁i , j ▁> ▁r ( n ). ▁To ▁compute ▁r , ▁compare ▁the ▁binary ▁expansion ▁of ▁x ▁with ▁the ▁binary ▁expansion ▁of ▁q i ▁for ▁larger ▁and ▁larger ▁values ▁of ▁i . ▁The ▁definition ▁of ▁q i ▁causes ▁a ▁single ▁binary ▁digit ▁to ▁go ▁from ▁ 0 ▁to ▁ 1 ▁each ▁time ▁i ▁increases ▁by ▁ 1 . ▁Thus ▁there ▁will ▁be ▁some ▁n ▁where ▁a ▁large ▁enough ▁initial ▁segment ▁of ▁x ▁has ▁already ▁been ▁determined ▁by ▁q n ▁that ▁no ▁additional ▁binary ▁digits ▁in ▁that ▁segment ▁could ▁ever ▁be ▁turned ▁on , ▁which ▁leads ▁to ▁an ▁estimate ▁on ▁the ▁distance ▁between ▁q i ▁and ▁q j ▁for ▁i , j ▁> ▁n . ▁▁ ▁If ▁any ▁such ▁a ▁function ▁r ▁were ▁comput able , ▁it ▁would ▁lead ▁to ▁a ▁decision ▁procedure ▁for ▁A , ▁as ▁follows . ▁Given ▁an ▁input ▁k , ▁compute ▁r (