reserve X for RealUnitarySpace,
  x, g, g1, h for Point of X,
  a, p, r, M, M1, M2 for Real,
  seq, seq1, seq2, seq3 for sequence of X,
  Nseq for increasing sequence of NAT,

  k, l, l1, l2, l3, n, m, m1, m2 for Nat;

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;
