reserve a, b, p, q for Real;

theorem Th9:
  for a,b,c be Real_Sequence st (for n be Nat holds a.n
<= b.n+c.n ) & b is convergent & c is convergent & a is non-decreasing holds a
  is convergent & lim a <= lim b + lim c
proof
  let a,b,c be Real_Sequence such that
A1: for n be Nat holds a.n <= b.n+c.n and
A2: b is convergent & c is convergent and
A3: a is non-decreasing;
A4: now
    let n be Nat;
    a.n <=b.n + c.n by A1;
    hence a.n <= (b+c).n by SEQ_1:7;
  end;
A5: b+c is convergent by A2;
  hence a is convergent by A3,A4,Th8;
  lim (b+c) = lim b + lim c by A2,SEQ_2:6;
  hence thesis by A3,A5,A4,Th8;
end;
