
theorem LMXFIN6:
  for d be XFinSequence of REAL,k be Nat
  st len d = 1 & d is nonnegative-yielding holds
  seq_p(d) in Big_Oh( seq_n^(k) )
  proof
    let d be XFinSequence of REAL,k be Nat;
    assume AS: len d = 1 & d is nonnegative-yielding;
    then consider a be Real such that
    P1: a = d.0 & for x be Nat holds (seq_p(d)).x = a by LMXFIN5;
    set y = seq_p(d);
    set c = a + 1;
    XA1:a + 0 < a + 1 by XREAL_1:8;
    0 in Segm 1 by NAT_1:44;then
    ASX: 0 <= d.0 by AS,FUNCT_1:3;
    A1: now
    let n be Element of NAT;
    assume A2: n >= 2;
    then A3: n > 1 by XXREAL_0:2;
    A4: (seq_n^ k) . n = n to_power k by A2,ASYMPT_1:def 3;
    1 <= ((seq_n^ k) . n)
    proof
      per cases;
      suppose k= 0;
        hence 1 <= ((seq_n^ k) . n) by A4,POWER:24;
      end;
      suppose 0 < k;
        hence 1 <= ((seq_n^ k) . n) by A4,A3,POWER:35;
      end;
    end; then
    1*a <= ((seq_n^ k) . n) * c by XA1,XREAL_1:66,P1,ASX;
    hence y.n <= c * ((seq_n^ k) . n) by P1;
    thus y.n >= 0 by P1,ASX;
  end;
  y is Element of Funcs (NAT,REAL) by FUNCT_2:8;
  hence y in Big_Oh (seq_n^ k) by A1,P1,ASX;
end;
