reserve X for RealNormSpace;

theorem Th6:
  for X be RealNormSpace, S be sequence of X, St be sequence of
  MetricSpaceNorm X st S = St & St is convergent holds lim St = lim S
proof
  let X be RealNormSpace, S be sequence of X, St be sequence of
  MetricSpaceNorm X;
  assume that
A1: S=St and
A2: St is convergent;
  reconsider xt=lim S as Point of MetricSpaceNorm X;
  S is convergent by A1,A2,Th5;
  then
  for r be Real st 0 < r
   ex m be Nat st for n be Nat
  st m <= n holds ||. S.n - lim S.|| < r by NORMSP_1:def 7;
  then St is_convergent_in_metrspace_to xt by A1,Th4;
  hence thesis by METRIC_6:11;
end;
