
theorem LMXFIN8:
  for k be Nat,a be Real,y be Real_Sequence
  st 0<=a &
  for i be Nat holds
  y.i = a* (i to_power k) holds y in Big_Oh( seq_n^k )
  proof
    let k be Nat,a be Real,y be Real_Sequence;
    assume AS: 0 <= a &
    for n be Nat holds y.n = a* (n to_power k);
    set c = a + 1;
    XA1:a + 0 < a + 1 by XREAL_1:8;
    A1: now
    let n be Element of NAT;
    assume A2: n >= 2;
    A4: (seq_n^ k) . n = n to_power k by A2,ASYMPT_1:def 3; then
    A5: y.n = a* ((seq_n^ k) . n ) by AS;
    hence y.n <= c * ((seq_n^ k) . n) by XA1,A4,XREAL_1:64;
    thus y.n >= 0 by A4,A5,AS;
  end;
  y is Element of Funcs (NAT,REAL) by FUNCT_2:8;
  hence y in Big_Oh (seq_n^ k) by AS,A1;
end;
