reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:COMPLEX,D:],D;
reserve B,C for Coset of W;
reserve CNS for ComplexNormSpace;
reserve x, y, w, g, g1, g2 for Point of CNS;
reserve S, S1, S2 for sequence of CNS;
reserve n, m, m1, m2 for Nat;
reserve r for Real;

theorem Th117:
  S is convergent implies ||.S.|| is convergent
proof
  assume S is convergent;
  then consider g such that
A1: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r;
  now
    let r be Real;
    assume
A2: 0 < r;
    consider m1 such that
A3: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,A2;
    take k = m1;
      let n;
      assume k <= n;
      then
A4:   ||.(S.n) - g.|| < r by A3;
      |.||.(S.n).|| - ||.g.||.| <= ||.(S.n) - g.|| by Th110;
      then |.||.(S.n).|| - ||.g.||.| < r by A4,XXREAL_0:2;
      hence |.||.S.||.n - ||.g.||.| < r by NORMSP_0:def 4;
  end;
  hence thesis by SEQ_2:def 6;
end;
