
theorem Th39:
  for F being sequence of ExtREAL, n being Nat st
  F is nonnegative holds (Ser F).n <= (Ser F).(n + 1) & 0. <= (Ser F).n
proof
  let F be sequence of ExtREAL, n be Nat;
  assume A0: F is nonnegative;
  set FF = Ser F;
  defpred P[Nat] means FF.$1 <= FF.($1 + 1) & 0. <= FF.$1;
    reconsider y = FF.0 as R_eal;
A1: FF.(0 + 1) = y + F.1 by Def11;
A2: for k being Nat st P[k] holds P[k + 1]
  proof
    let k be Nat;
    assume that
A3: FF.k <= FF.(k + 1) and
A4: 0. <= FF.k;
    FF.((k+1) + 1) = FF.(k + 1) + F.((k+1) + 1) by Def11;
    hence thesis by A3,A4,Th38,A0,XXREAL_3:39;
  end;
  FF.0 = F.0 by Def11; then
A6: P[0] by A1,XXREAL_3:39,A0,Th38;
  for n being Nat holds P[n] from NAT_1:sch 2(A6,A2);
  hence thesis;
end;
