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 Th12:
  (for n holds ||. seq.n .|| <= rseq.n) & rseq is convergent & lim
  (rseq)=0 implies seq is convergent & lim(seq)=0.X
proof
  assume that
A1: for n holds ||. seq.n .|| <= rseq.n and
A2: rseq is convergent and
A3: lim(rseq)=0;
   now
    let p be Real;
    assume 0<p;
    then consider n being Nat such that
A4: for m being Nat st n<=m holds |.rseq.m-0 .|<p by A2,A3,SEQ_2:def 7;
      reconsider n as Nat;
     take n;
      let m;
      assume n<=m;
      then
A5:   |.rseq.m-0 .|<p by A4;
A6:   ||. seq.m-0.X .|| = ||. seq.m .|| by RLVECT_1:13;
A7:   rseq.m <= |.rseq.m.| by ABSVALUE:4;
      ||. seq.m .|| <= rseq.m by A1;
      then ||. seq.m - 0.X .|| <= |.rseq.m.| by A6,A7,XXREAL_0:2;
      hence ||. seq.m - 0.X .|| < p by A5,XXREAL_0:2;
  end;
  then
A8: for p be Real st 0<p
   ex n st for m st n<=m holds ||. seq.m - 0.X .|| < p;
  hence seq is convergent;
  hence thesis by A8,NORMSP_1:def 7;
end;
