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
  S is convergent & lim S = g implies ||.S - g.|| is convergent & lim
  ||.S - g.|| = 0
proof
  assume that
A1: S is convergent and
A2: lim S = g;
A3: now
    let r be Real;
    assume
A4: 0 < r;
    consider m1 such that
A5: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,A2,A4,Def16;
    take k = m1;
    let n;
    assume k <= n;
    then ||.(S.n) - g.|| < r by A5;
    then
A6: ||.((S.n) - g ) - 09(CNS).|| < r by RLVECT_1:13;
    |.||.(S.n) - g.|| - ||.09(CNS).||.| <= ||.((S.n) - g ) - 09(CNS).||
    by Th110;
    then |.||.(S.n) - g.|| - ||.09(CNS).||.| < r by A6,XXREAL_0:2;
    then |.(||.(S.n) - g.|| - 0).| < r;
    then |.(||.(S - g).n.|| - 0).| < r by NORMSP_1:def 4;
    hence |.(||.S - g.||.n - 0).| < r by NORMSP_0:def 4;
  end;
  ||.S - g.|| is convergent by A1,Th115,Th117;
  hence thesis by A3,SEQ_2:def 7;
end;
