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