reserve c, c1, d for Real,
  k for Nat,
  n, m, N, n1, N1, N2, N3, N4, N5, M for Element of NAT,
  x for set;

theorem Th4:
  for f,g being Real_Sequence st f is convergent & g is convergent
  & g majorizes f holds lim(f) <= lim(g)
proof
  let f,g be Real_Sequence;
  assume that
A1: f is convergent & g is convergent and
A2: ex N st for n st n >= N holds f.n <= g.n;
  consider N such that
A3: for n st n >= N holds f.n <= g.n by A2;
  now
    let n be Nat;
A4:  n in NAT by ORDINAL1:def 12;
    assume n >= N;
    then f.n <= g.n by A3,A4;
    then
A5: f.n-f.n<=g.n-f.n by XREAL_1:9;
    (g-f).n= g.n+(-f).n by SEQ_1:7
      .=g.n+-f.n by SEQ_1:10
      .= g.n-f.n;
    hence 0<=(g-f).n by A5;
  end;
  then
A6: g-f is eventually-nonnegative;
A7: lim(g-f) = (lim(g)) - (lim(f)) by A1,SEQ_2:12;
  g-f is convergent by A1,SEQ_2:11;
  then 0 <= lim(g-f) by A6,Th3;
  then 0 + (lim(f)) <= (lim(g)) - (lim(f)) + (lim(f)) by A7,XREAL_1:6;
  hence thesis;
end;
