reserve a, b, p, q for Real;

theorem Th8:
  for a,b be Real_Sequence st (for n be Nat holds a.n <=
b.n ) & b is convergent & a is non-decreasing holds a is convergent & lim a <=
  lim b
proof
  let a,b be Real_Sequence such that
A1: for n be Nat holds a.n <= b.n and
A2: b is convergent and
A3: a is non-decreasing;
A4: b is bounded by A2;
A5: a is bounded_above
  proof
    consider r be Real such that
A6: for n be Nat holds b.n < r by A4,SEQ_2:def 3;
    now
      let n be Nat;
      a.n <=b.n by A1;
      then a.n <= r by A6,XXREAL_0:2;
      then a.n+0 < r+1 by XREAL_1:8;
      hence a.n < r+1;
    end;
    hence thesis by SEQ_2:def 3;
  end;
  hence a is convergent by A3;
  thus thesis by A1,A2,A3,A5,SEQ_2:18;
end;
