reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th20:
  for X be RealNormSpace for seq be sequence of X for g be Point
  of X holds seq is convergent & lim seq = g implies ||.seq.|| is convergent &
  lim ||.seq.|| = ||.g.||
proof
  let X be RealNormSpace;
  let seq be sequence of X;
  let g be Point of X;
  assume that
A1: seq is convergent and
A2: lim seq = g;
A3: now
    let r be Real;
    assume
A4: r > 0;
    consider m1 be Nat such that
A5: for n be Nat st n >= m1 holds ||.(seq.n) - g.|| < r by A1,A2,A4,
NORMSP_1:def 7;
     reconsider k = m1 as Nat;
    take k;
    now
      let n be Nat;
      assume n >= k;
      then
A6:   ||.(seq.n) - g.|| < r by A5;
      |.||.(seq.n).|| - ||.g.||.| <= ||.(seq.n) - g.|| by NORMSP_1:9;
      then |.||.(seq.n).|| - ||.g.||.| < r by A6,XXREAL_0:2;
      hence |.||.seq.||.n - ||.g.||.| < r by NORMSP_0:def 4;
    end;
    hence for n be Nat st k <= n holds |.||.seq.||.n - ||.g.||.| <
    r;
  end;
  ||.seq.|| is convergent by A1,NORMSP_1:23;
  hence thesis by A3,SEQ_2:def 7;
end;
