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 Th9:
  seq is convergent iff ex g st for r st r > 0 ex m st for n st n
  >= m holds ||.(seq.n) - g.|| < r
proof
  thus seq is convergent implies ex g st for r st r > 0 ex m st for n st n >=
  m holds ||.(seq.n) - g.|| < r
  proof
    assume seq is convergent;
    then consider g such that
A1: for r st r > 0 ex m st for n st n >= m holds dist((seq.n) , g) < r;
    take g;
    let r;
    assume r > 0;
    then consider m1 such that
A2: for n st n >= m1 holds dist((seq.n) , g) < r by A1;
    take k = m1;
    let n;
    assume n >= k;
    then dist((seq.n) , g) < r by A2;
    hence thesis by BHSP_1:def 5;
  end;
  ( ex g st for r st r > 0 ex m st for n st n >= m holds ||.(seq.n) - g.||
  < r ) implies seq is convergent
  proof
    given g such that
A3: for r st r > 0 ex m st for n st n >= m holds ||.(seq.n) - g.|| < r;
    take g;
    let r;
    assume r > 0;
    then consider m1 such that
A4: for n st n >= m1 holds ||.(seq.n) - g.|| < r by A3;
    take k = m1;
    let n;
    assume n >= k;
    then ||.(seq.n) - g.|| < r by A4;
    hence thesis by BHSP_1:def 5;
  end;
  hence thesis;
end;
