let a, b be Nat; :: thesis: ( a < b implies seq_n^ a in Big_Oh (seq_n^ b) )
assume AS: a < b ; :: thesis: seq_n^ a in Big_Oh (seq_n^ b)
set g = seq_n^ b;
set f = seq_n^ a;

LL11: now :: thesis: for n being Element of NAT st n >= 2 holds
( (seq_n^ a) . n <= 1 * ((seq_n^ b) . n) & (seq_n^ a) . n >= 0 )

let n be Element of NAT ; :: thesis: ( n >= 2 implies ( (seq_n^ a) . n <= 1 * ((seq_n^ b) . n) & (seq_n^ a) . n >= 0 ) )
assume A2: n >= 2 ; :: thesis: ( (seq_n^ a) . n <= 1 * ((seq_n^ b) . n) & (seq_n^ a) . n >= 0 )
then A3: n > 1 by XXREAL_0:2;
A4: (seq_n^ a) . n = n to_power a by A2, ASYMPT_1:def 3;
(seq_n^ b) . n = n to_power b by A2, ASYMPT_1:def 3;
hence (seq_n^ a) . n <= 1 * ((seq_n^ b) . n) by AS, A3, A4, POWER:39; :: thesis: (seq_n^ a) . n >= 0
thus (seq_n^ a) . n >= 0 by A4; :: thesis: verum

end;

reconsider f = seq_n^ a as Element of Funcs (NAT,REAL) by FUNCT_2:8;
reconsider g = seq_n^ b as eventually-nonnegative Real_Sequence ;
f in Big_Oh g by LL11;
hence seq_n^ a in Big_Oh (seq_n^ b) ; :: thesis: verum