reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;

theorem Th24:
  seq is convergent & lim seq = g implies dist(seq, g) is
  convergent & lim dist(seq, g) = 0
proof
  assume
A1: seq is convergent & lim seq = g;
A2: now
    let r be Real;
    assume
A3: r > 0;
    consider m1 such that
A4: for n st n >= m1 holds dist((seq.n) , g) < r by A1,A3,Def2;
     reconsider k = m1 as Nat;
    take k;
    let n be Nat;
    dist((seq.n) , g) >= 0 by BHSP_1:37;
    then
A5: |.(dist((seq.n) , g) - 0).| = dist((seq.n) , g) by ABSVALUE:def 1;
    assume n >= k;
    then dist((seq.n) , g) < r by A4;
    hence |.(dist(seq, g).n - 0).| < r by A5,Def4;
  end;
  dist(seq, g) is convergent by A1,Th23;
  hence thesis by A2,SEQ_2:def 7;
end;
