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 Th23:
  seq is convergent & lim seq = g implies dist(seq, g) is convergent
proof
  assume
A1: seq is convergent & lim seq = g;
  now
    let r be Real;
    assume
A2: r > 0;
    consider m1 such that
A3: for n st n >= m1 holds dist((seq.n) , g) < r by A1,A2,Def2;
     reconsider k = m1 as Nat;
    take k;
    now
      let n be Nat;
      dist((seq.n) , g) >= 0 by BHSP_1:37;
      then
A4:   |.(dist((seq.n) , g) - 0).| = dist((seq.n) , g) by ABSVALUE:def 1;
      assume n >= k;
      then dist((seq.n) , g) < r by A3;
      hence |.(dist(seq, g).n - 0).| < r by A4,Def4;
    end;
    hence for n being Nat st k <= n holds |.(dist(seq, g).n - 0).| < r;
  end;
  hence thesis by SEQ_2:def 6;
end;
