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

theorem Th4:
  for X be RealUnitarySpace, S be sequence of X,
      St be sequence of MetricSpaceNorm RUSp2RNSp X,
      x be Point of X,
      xt be Point of MetricSpaceNorm RUSp2RNSp X st S = St
  & x = xt holds St is_convergent_in_metrspace_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 RealUnitarySpace, S be sequence of X,
  St be sequence of MetricSpaceNorm RUSp2RNSp X,
  x be Point of X, xt be Point of MetricSpaceNorm RUSp2RNSp X;
  assume
A1: S=St & x=xt;
  hereby
    assume
A5: St is_convergent_in_metrspace_to xt;
    let r be Real;
    assume 0 < r;
    then consider m be Nat such that
A6: for n being Nat st m <= n holds dist(St.n,xt) < r by A5;
    take m;
    let n be Nat;
    assume m <= n;
    then dist(St.n,xt) < r by A6;
    hence ||. S.n - x.|| < r by A1,Th1;
  end;
    assume
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;
    now
      let r be Real;
      assume r>0;
      then consider m be Nat such that
A4:   for n be Nat st m <= n holds ||. S.n - x.|| < r by A3;
      take m;
      let n be Nat;
      assume m <= n;
      then ||. S.n - x.|| < r by A4;
      hence dist(St.n,xt) < r by A1,Th1;
    end;
    hence St is_convergent_in_metrspace_to xt;
end;
