
theorem
  for f be eventually-nonnegative Real_Sequence,
      k be Nat st
  f in Big_Oh(seq_n^(k)) holds
  ex N be Nat st
  for n be Nat st N <= n holds f.n <= (seq_n^(k+1)).n
  proof
    let f be eventually-nonnegative Real_Sequence,
    k be Nat;
    assume f in Big_Oh(seq_n^(k));
    then consider t be Element of Funcs (NAT,REAL) such that
    L1: f=t &
    ex c being Real,N being Element of NAT st c > 0 &
    for n being Element of NAT st n >= N holds
    ( t . n <= c * ((seq_n^k).n) & t . n >= 0 );
    consider c being Real,N being Element of NAT such that L2: c > 0 &
    for n being Element of NAT st n >= N holds
    f . n <= c * ((seq_n^k).n) & f . n >= 0 by L1;
    set n = [/ max(N,c) \];
    P1:N <= max(N,c) & c <= max(N,c) by XXREAL_0:25;
    P2P:max(N,c) <= n by INT_1:def 7;then
    P2:N <= n & c <= n by P1, XXREAL_0:2;
    reconsider n as Element of NAT by INT_1:3,P2P,P1;
    take n;
    let x be Nat;
    A: x in NAT by ORDINAL1:def 12;
    assume P4:n <= x;then
    P4r:0 < x & N <= x & c <= x by P2,L2,XXREAL_0:2;
    P5: (seq_n^(k+1)).x = x *((seq_n^(k)).x) by TPOWSUCC,P2P,L2,P4,P1;
    P6: f.x <= c * ((seq_n^k).x) & f.x >= 0 by A,P4r,L2;
    (seq_n^(k)).x = x to_power k by P4,P2P,L2,A,ASYMPT_1:def 3,P1; then
    c*((seq_n^(k)).x) <= x*((seq_n^(k)).x) by P4r,XREAL_1:64;
    hence thesis by P5,XXREAL_0:2,P6;
  end;
