reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem Th29:
  s is convergent & s1 is bounded & (lim s)=0c implies s(#)s1 is convergent
proof
  assume that
A1: s is convergent and
A2: s1 is bounded and
A3: (lim s) = 0c;
  take g=0c;
  consider R such that
A4: 0<R and
A5: for m holds |.s1.m.|<R by A2,Th8;
  let p be Real such that
A6: 0<p;
A7: 0<p/R by A6,A4;
  then consider n1 such that
A8: for m st n1<=m holds |.s.m-0c.|<p/R by A1,A3,Def6;
  take n=n1;
  let m such that
A9: n<=m;
A10: |.((s(#)s1).m)-g.|=|.s.m*s1.m.| by VALUED_1:5
    .=|.s.m.|*|.s1.m.| by COMPLEX1:65;
  |.s.m.|=|.s.m-0c.|;
  then
A11: |.s.m.|<p/R by A8,A9;
  now
    (p/R)*R=p*R"*R by XCMPLX_0:def 9
      .=p*(R"*R)
      .=p*1 by A4,XCMPLX_0:def 7
      .=p;
    then
A12: (p/R)*|.s1.m.|<p by A5,A7,XREAL_1:68;
A13: 0<=|.s1.m.| by COMPLEX1:46;
    assume |.s1.m.|<>0;
    then |.((s(#)s1).m)-g.|<(p/R)*|.s1.m.| by A11,A10,A13,XREAL_1:68;
    hence thesis by A12,XXREAL_0:2;
  end;
  hence thesis by A6,A10;
end;
