reserve X for RealNormSpace;

theorem
  for X be RealNormSpace, S be sequence of X, St be sequence of
LinearTopSpaceNorm X, x be Point of X, xt be Point of LinearTopSpaceNorm X st S
  =St & x=xt holds St is_convergent_to xt iff
 for r be Real st 0 < r ex m be
  Nat st for n be Nat st m <= n holds ||.(S.n) - x.|| < r
proof
  let X be RealNormSpace, S be sequence of X, St be sequence of
  LinearTopSpaceNorm X, x be Point of X, xt be Point of LinearTopSpaceNorm X;
  reconsider xxt=xt as Point of TopSpaceNorm X by Def4;
  assume
A1: S=St & x=xt;
  the carrier of LinearTopSpaceNorm X = the carrier of TopSpaceNorm X by Def4;
  then reconsider SSt = St as sequence of TopSpaceNorm X;
  St is_convergent_to xt iff SSt is_convergent_to xxt by Th26;
  hence thesis by A1,Th12;
end;
