reserve X for RealNormSpace;

theorem Th4:
  for X be RealNormSpace, S be sequence of X, St be sequence of
MetricSpaceNorm X, x be Point of X, xt be Point of MetricSpaceNorm 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 RealNormSpace, S be sequence of X, St be sequence of
  MetricSpaceNorm X, x be Point of X, xt be Point of MetricSpaceNorm X;
  assume
A1: S=St & x=xt;
A2: now
    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,Def1;
    end;
    hence St is_convergent_in_metrspace_to xt by METRIC_6:def 2;
  end;
  now
    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,
METRIC_6:def 2;
    take m;
    let n be Nat;
    assume m <= n;
    then dist(St.n,xt) < r by A6;
    hence ||. S.n - x.|| < r by A1,Def1;
  end;
  hence thesis by A2;
end;
