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 Th30:
  s is convergent & s1 is bounded & (lim s)=0c implies lim(s(#)s1) =0c
proof
  assume that
A1: s is convergent and
A2: s1 is bounded and
A3: lim s=0c;
A4: now
    consider R such that
A5: 0<R and
A6: for m holds |.s1.m.|<R by A2,Th8;
    let p be Real such that
A7: 0<p;
A8: 0<p/R by A7,A5;
    then consider n1 such that
A9: for m st n1<=m holds |.s.m-0c.|<p/R by A1,A3,Def6;
    take n=n1;
    let m;
    (p/R)*R=p*R"*R by XCMPLX_0:def 9
      .=p*(R"*R)
      .=p*1 by A5,XCMPLX_0:def 7;
    then
A10: (p/R)*|.s1.m.|<p by A6,A8,XREAL_1:68;
    assume n<=m;
    then
A11: |.s.m-0c.|<p/R by A9;
A12: |.((s(#)s1).m)-0c.|=|.s.m*s1.m.| by VALUED_1:5
      .=|.s.m.|*|.s1.m.| by COMPLEX1:65;
A13: 0<=|.s1.m.| by COMPLEX1:46;
    now
      assume |.s1.m.|<>0;
      then |.((s(#)s1).m)-0c.|<(p/R)*|.s1.m.| by A11,A12,A13,XREAL_1:68;
      hence |.((s(#)s1).m)-0c.|<p by A10,XXREAL_0:2;
    end;
    hence |.((s(#)s1).m)-0c.|<p by A7,A12;
  end;
  s(#)s1 is convergent by A1,A2,A3,Th29;
  hence thesis by A4,Def6;
end;
