reserve n, m for Element of NAT;
reserve z for Complex;

theorem Th4:
  for X be RealNormSpace, seq be sequence of X holds seq is
convergent & lim seq = 0.X iff ||. seq .|| is convergent & lim ||. seq .|| = 0
proof
  let X be RealNormSpace, seq be sequence of X;
  thus seq is convergent & lim seq = 0.X implies ||. seq .|| is convergent &
  lim ||. seq .|| = 0
  proof
    assume
A1: seq is convergent & lim seq = 0.X;
    then lim ||. seq .|| = ||.0.X.|| by LOPBAN_1:20;
    hence thesis by A1,LOPBAN_1:20,NORMSP_1:1;
  end;
  assume
A2: ||. seq .|| is convergent & lim ||. seq .|| = 0;
A3: now
    let p be Real;
    assume 0 < p;
    then consider m be Nat such that
A4: for n be Nat st m <= n holds |.||. seq .||.n - 0 .| < p
    by A2,SEQ_2:def 7;
    take m;
    hereby
      let n be Nat;
      assume m <= n;
      then |.||. seq .||.n - 0 .| < p by A4;
      then
A5:   |. ||. seq.n .|| .| < p by NORMSP_0:def 4;
      0 <= ||. seq.n .|| by NORMSP_1:4;
      then ||. seq.n .|| < p by A5,ABSVALUE:def 1;
      hence ||.seq.n-0.X.|| < p by RLVECT_1:13;
    end;
  end;
  then seq is convergent by NORMSP_1:def 6;
  hence thesis by A3,NORMSP_1:def 7;
end;
