 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem Th5:
  for X be RealUnitarySpace, S be sequence of X,
      St be sequence of MetricSpaceNorm RUSp2RNSp X st S = St
    holds St is convergent iff S is convergent
proof
  let X be RealUnitarySpace, S be sequence of X,
      St be sequence of MetricSpaceNorm RUSp2RNSp X;
  assume
A1: S=St;
  hereby
    assume St is convergent;
    then consider xt be Point of MetricSpaceNorm RUSp2RNSp X such that
A4: St is_convergent_in_metrspace_to xt by METRIC_6:10;
    reconsider x=xt as Point of X;
    for r be Real st 0 < r ex m be Nat st
      for n be Nat st m <= n holds ||. S.n - x.|| < r by A1,A4,Th4;
    hence S is convergent by BHSP_2:9;
  end;
    assume S is convergent;
    then consider x be Point of X such that
A3: for r be Real st 0 < r ex m be Nat st
     for n be Nat st m <= n holds ||. S.n - x.|| < r by BHSP_2:9;
    reconsider xt=x as Point of MetricSpaceNorm RUSp2RNSp X;
    St is_convergent_in_metrspace_to xt by A1,A3,Th4;
    hence St is convergent;
end;
