reserve n,n1,n2,m for Nat,
  r,r1,r2,p,g1,g2,g for Real,
  seq,seq9,seq1 for Real_Sequence,
  y for set;

theorem Th25:
  seq is convergent & seq1 is bounded & lim seq=0 implies
  seq(#)seq1 is convergent
proof
  assume that
A1: seq is convergent and
A2: seq1 is bounded and
A3: lim seq=0;
  reconsider g1=0 as Real;
  take g=g1;
  let p such that
A4: 0<p;
  consider r such that
A5: 0<r and
A6: for m holds |.seq1.m.|<r by A2,Th3;
A7: 0<p/r by A4,A5;
  then consider n1 such that
A8: for m st n1<=m holds |.seq.m-0.|<p/r by A1,A3,Def6;
  take n=n1;
  let m such that
A9: n<=m;
  |.seq.m.|=|.seq.m-0.|;
  then
A10: |.seq.m.|<p/r by A8,A9;
A11: |.((seq(#)seq1).m)-g.|=|.seq.m*seq1.m-0.| by SEQ_1:8
    .=|.seq.m.|*|.seq1.m.| by COMPLEX1:65;
  now
    assume
A12: |.seq1.m.|<>0;
    (p/r)*r=p*r"*r by XCMPLX_0:def 9
      .=p*(r"*r)
      .=p*1 by A5,XCMPLX_0:def 7
      .=p;
    then
A13: (p/r)*|.seq1.m.|<p by A6,A7,XREAL_1:68;
    0<=|.seq1.m.| by COMPLEX1:46;
    then |.((seq(#)seq1).m)-g.|<(p/r)*|.seq1.m.| by A10,A11,A12,XREAL_1:68;
    hence thesis by A13,XXREAL_0:2;
  end;
  hence thesis by A4,A11;
end;
