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 Th21:
  seq is convergent & lim seq = g implies ||.seq.|| is convergent
  & lim ||.seq.|| = ||.g.||
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;
    now
      let n be Nat;
      assume n >= k;
      then
A6:   ||.(seq.n) - g.|| < r by A5;
      |.||.(seq.n).|| - ||.g.||.| <= ||.(seq.n) - g.|| by BHSP_1:33;
      then |.||.(seq.n).|| - ||.g.||.| < r by A6,XXREAL_0:2;
      hence |.||.seq.||.n - ||.g.||.| < r by Def3;
    end;
    hence for n being Nat st k <= n holds |.||.seq.||.n - ||.g.||.| < r;
  end;
  ||.seq.|| is convergent by A1,Th20;
  hence thesis by A3,SEQ_2:def 7;
end;
