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 Th19:
  seq is convergent implies ( lim seq = g iff for r st r > 0 ex m
  st for n st n >= m holds ||.(seq.n) - g.|| < r )
proof
  assume
A1: seq is convergent;
  thus lim seq = g implies for r st r > 0 ex m st for n st n >= m holds ||.(
  seq.n) - g.|| < r
  proof
    assume
A2: lim seq = g;
    let r;
    assume r > 0;
    then consider m1 such that
A3: for n st n >= m1 holds dist((seq.n) , g) < r by A1,A2,Def2;
    take k = m1;
    let n;
    assume n >= k;
    then dist((seq.n) , g) < r by A3;
    hence thesis by BHSP_1:def 5;
  end;
  ( for r st r > 0 ex m st for n st n >= m holds ||.(seq.n) - g.|| < r )
  implies lim seq = g
  proof
    assume
A4: for r st r > 0 ex m st for n st n >= m holds ||.(seq.n) - g.|| < r;
    now
      let r;
      assume r > 0;
      then consider m1 such that
A5:   for n st n >= m1 holds ||.(seq.n) - g.|| < r by A4;
      take k = m1;
      let n;
      assume n >= k;
      then ||.(seq.n) - g.|| < r by A5;
      hence dist((seq.n) , g) < r by BHSP_1:def 5;
    end;
    hence thesis by A1,Def2;
  end;
  hence thesis;
end;
