let seq, seq1 be Real_Sequence; :: thesis: ( seq is convergent & seq1 is bounded & lim seq = 0 implies lim (seq (#) seq1) = 0 )
assume that
A1: seq is convergent and
A2: seq1 is bounded and
A3: lim seq = 0 ; :: thesis: lim (seq (#) seq1) = 0
A4: seq (#) seq1 is convergent by A1, A2, A3, Th39;

now

let p be real number ; :: thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((seq (#) seq1) . m) - 0) < p )
assume A5: 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs (((seq (#) seq1) . m) - 0) < p
consider r being real number such that
A6: 0 < r and
A7: for m being Element of NAT holds abs (seq1 . m) < r by A2, Th15;
A8: 0 < p / r by A5, A6, XREAL_1:141;
then consider n1 being Element of NAT such that
A9: for m being Element of NAT st n1 <= m holds
abs ((seq . m) - 0) < p / r by A1, A3, Def7;
take n = n1; :: thesis: for m being Element of NAT st n <= m holds
abs (((seq (#) seq1) . m) - 0) < p
let m be Element of NAT ; :: thesis: ( n <= m implies abs (((seq (#) seq1) . m) - 0) < p )
assume n <= m ; :: thesis: abs (((seq (#) seq1) . m) - 0) < p
then A10: abs ((seq . m) - 0) < p / r by A9;
A11: abs (((seq (#) seq1) . m) - 0) = abs (((seq . m) * (seq1 . m)) - 0) by SEQ_1:12
.= (abs (seq . m)) * (abs (seq1 . m)) by COMPLEX1:151 ;

now

assume A12: abs (seq1 . m) <> 0 ; :: thesis: abs (((seq (#) seq1) . m) - 0) < p
(p / r) * r = (p * (r ")) * r by XCMPLX_0:def 9
.= p * ((r ") * r)
.= p * 1 by A6, XCMPLX_0:def 7 ;
then A13: (p / r) * (abs (seq1 . m)) < p by A7, A8, XREAL_1:70;
0 <= abs (seq1 . m) by COMPLEX1:132;
then abs (((seq (#) seq1) . m) - 0) < (p / r) * (abs (seq1 . m)) by A10, A11, A12, XREAL_1:70;
hence abs (((seq (#) seq1) . m) - 0) < p by A13, XXREAL_0:2; :: thesis: verum

end;

hence abs (((seq (#) seq1) . m) - 0) < p by A5, A11; :: thesis: verum

end;

hence lim (seq (#) seq1) = 0 by A4, Def7; :: thesis: verum