let seq1, seq2 be Real_Sequence; :: thesis: ( seq1 is divergent_to+infty & ex r being Real st
( r > 0 & ( for n being Element of NAT holds seq2 . n >= r ) ) implies seq1 (#) seq2 is divergent_to+infty )
assume that
A1: seq1 is divergent_to+infty and
A2: ex r being Real st
( r > 0 & ( for n being Element of NAT holds seq2 . n >= r ) ) ; :: thesis: seq1 (#) seq2 is divergent_to+infty
consider M being Real such that
A3: M > 0 and
A4: for n being Element of NAT holds seq2 . n >= M by A2;
let r be Real; :: according to LIMFUNC1:def 4 :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
r < (seq1 (#) seq2) . m
A5: 0 <= abs r by COMPLEX1:132;
consider n being Element of NAT such that
A6: for m being Element of NAT st n <= m holds
(abs r) / M < seq1 . m by A1, Def4;
take n ; :: thesis: for m being Element of NAT st n <= m holds
r < (seq1 (#) seq2) . m
let m be Element of NAT ; :: thesis: ( n <= m implies r < (seq1 (#) seq2) . m )
assume n <= m ; :: thesis: r < (seq1 (#) seq2) . m
then (abs r) / M < seq1 . m by A6;
then ((abs r) / M) * M < (seq1 . m) * (seq2 . m) by A3, A4, A5, XREAL_1:99;
then A7: abs r < (seq1 . m) * (seq2 . m) by A3, XCMPLX_1:88;
r <= abs r by ABSVALUE:11;
then r < (seq1 . m) * (seq2 . m) by A7, XXREAL_0:2;
hence r < (seq1 (#) seq2) . m by SEQ_1:12; :: thesis: verum