reserve a, b for Real;
reserve RNS for RealNormSpace;
reserve x, y, z, g, g1, g2 for Point of RNS;
reserve S, S1, S2 for sequence of RNS;
reserve k, n, m, m1, m2 for Nat;
reserve r for Real;
reserve f for Function;
reserve d, s, t for set;

theorem Th23:
  S is convergent implies ||.S.|| is convergent
proof
  assume S is convergent;
  then consider g such that
A1: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r;
  now
    let r be Real;
    assume
A2: 0 < r;
    consider m1 such that
A3: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,A2;
     reconsider k = m1 as Nat;
    take k;
    now
      let n be Nat;
      assume
A4:     k <= n;
       reconsider nn = n as Nat;
A5:   ||.(S.nn) - g.|| < r by A3,A4;
      |. ||.(S.nn).|| - ||.g.|| .| <= ||.(S.nn) - g.|| by Th9;
      then |. ||.(S.nn).|| - ||.g.|| .| < r by A5,XXREAL_0:2;
      hence |. ||.S.||.n qua Real - ||.g.|| .| < r by NORMSP_0:def 4;
    end;
    hence for n being Nat st k <= n holds |. ||.S.||.n - ||.g.|| .| < r;
  end;
  hence thesis by SEQ_2:def 6;
end;
