
theorem Th2:
  for F being sequence of ExtREAL st F is nonnegative holds for
  n being Nat holds F.n <= Ser(F).n
proof
  let F be sequence of ExtREAL;
  assume
A1: F is nonnegative;
  let n be Nat;
  per cases;
  suppose
    n = 0;
    hence thesis by SUPINF_2:def 11;
  end;
  suppose
    n <> 0;
    then consider k being Nat such that
A2: n = k + 1 by NAT_1:6;
    reconsider k as Element of NAT by ORDINAL1:def 12;
A3: 0. <= Ser(F).k by A1,SUPINF_2:40;
    Ser(F).n = Ser(F).k + F.n by A2,SUPINF_2:def 11;
    then 0. + F.n <= Ser(F).n by A3,XXREAL_3:36;
    hence thesis by XXREAL_3:4;
  end;
end;
