:: deftheorem defines Big_Oh ASYMPT_0:def 12 :

for f being eventually-nonnegative Real_Sequence

for X being set holds Big_Oh (f,X) = { t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st

( c > 0 & ( for n being Element of NAT st n >= N & n in X holds

( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;

for f being eventually-nonnegative Real_Sequence

for X being set holds Big_Oh (f,X) = { t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st

( c > 0 & ( for n being Element of NAT st n >= N & n in X holds

( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;