theorem
  seq1 is convergent & lim seq1 = g & seq1 is_compared_to seq2 implies
  seq2 is convergent & lim seq2 = g
proof
  assume that
A1: seq1 is convergent & lim seq1 = g and
A2: seq1 is_compared_to seq2;
A3: now
    let r;
    assume r > 0;
    then
A4: r/2 > 0 by XREAL_1:215;
    then consider m1 being Nat such that
A5: for n being Nat st n >= m1 holds dist((seq1.n), g) < r/2
           by A1,BHSP_2:def 2;
    consider m2 such that
A6: for n st n >= m2 holds dist((seq1.n), (seq2.n)) < r/2 by A2,A4;
    reconsider m = m1 + m2 as Nat;
    take m;
    let n being Nat such that
A7: n >= m;
    m >= m2 by NAT_1:12;
    then n >= m2 by A7,XXREAL_0:2;
    then
A8: dist((seq1.n), (seq2.n)) < r/2 by A6;
A9: dist((seq2.n), g) <= dist((seq2.n), (seq1.n)) + dist((seq1.n), g) by
BHSP_1:35;
    m1 + m2 >= m1 by NAT_1:12;
    then n >= m1 by A7,XXREAL_0:2;
    then dist((seq1.n), g) < r/2 by A5;
    then dist((seq2.n), (seq1.n)) + dist((seq1.n), g) < r/2 + r/2 by A8,
XREAL_1:8;
    hence dist((seq2.n), g) < r by A9,XXREAL_0:2;
  end;
  then seq2 is convergent by BHSP_2:def 1;
  hence thesis by A3,BHSP_2:def 2;
end;
