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 Th22:
  seq is convergent & lim seq = g implies ||.seq - g.|| is
  convergent & lim ||.seq - g.|| = 0
proof
  assume that
A1: seq is convergent and
A2: lim seq = g;
A3: now
    let r be Real;
    assume
A4: r > 0;
    consider m1 such that
A5: for n st n >= m1 holds ||.(seq.n) - g.|| < r by A1,A2,A4,Th19;
     reconsider k = m1 as Nat;
    take k;
    let n be Nat;
    assume n >= k;
    then ||.(seq.n) - g.|| < r by A5;
    then
A6: ||.((seq.n) - g ) - 09(X).|| < r by RLVECT_1:13;
    |.||.(seq.n) - g.|| - ||.09(X).||.| <= ||.((seq.n) - g ) - 09( X )
    .|| by BHSP_1:33;
    then |.||.(seq.n) - g.|| - ||.09(X).||.| < r by A6,XXREAL_0:2;
    then |.(||.(seq.n) - g.|| - 0).| < r by BHSP_1:26;
    then |.(||.(seq - g).n.|| - 0).| < r by NORMSP_1:def 4;
    hence |.(||.seq - g.||.n - 0).| < r by Def3;
  end;
  ||.seq - g.|| is convergent by A1,Th8,Th20;
  hence thesis by A3,SEQ_2:def 7;
end;
