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
  S is convergent & lim S = g implies ||.S - g.|| is convergent & lim
  ||.S - g.|| = 0
proof
  assume that
A1: S is convergent and
A2: lim S = g;
A3: now
    let r be Real;
    assume
A4: 0 < r;
    consider m1 such that
A5: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,A2,A4,Def7;
     reconsider k = m1 as Nat;
    take k;
    let n be Nat;
    assume
A6:    k <= n;
     reconsider nn=n as Nat;
    ||.(S.nn) - g.|| < r by A5,A6;
    then
A7: ||.((S.nn) - g ) - 09(RNS).|| < r;
    |.||.(S.nn) - g.|| - ||.09(RNS).||.| <= ||.((S.nn) - g ) - 09(RNS).||
    by Th9;
    then |.||.(S.nn) - g.|| - ||.09(RNS).||.| < r by A7,XXREAL_0:2;
    then |.(||.(S - g).nn.|| - 0).| < r by Def4;
    hence |.(||.S - g.||.n - 0).| < r by NORMSP_0:def 4;
  end;
  ||.S - g.|| is convergent by A1,Th21,Th23;
  hence thesis by A3,SEQ_2:def 7;
end;
