reserve X for Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th2:
  for z st for n being Nat holds s.n = z holds lim s = z
proof
  let z;
  assume
A1: for n being Nat holds s.n = z;
A2: now
    let p be Real such that
A3: 0<p;
     reconsider k = 0 as Nat;
    take k;
    let n such that
    k<=n;
    s.n = z by A1;
    hence ||.s.n - z.|| < p by A3,NORMSP_1:6;
  end;
  then s is convergent;
  hence thesis by A2,NORMSP_1:def 7;
end;
