let seq be Complex_Sequence; :: thesis: ( ( for n being Element of NAT holds seq . n = 0c ) implies seq is absolutely_summable )
assume A1: for n being Element of NAT holds seq . n = 0c ; :: thesis: seq is absolutely_summable
take 0 ; :: according to SEQ_2:def 6,SERIES_1:def 2,COMSEQ_3:def 11 :: thesis: for b₁ being set holds
( b₁ <= 0 or ex b₂ being Element of NAT st
for b₃ being Element of NAT holds
( not b₂ <= b₃ or not b₁ <= abs (((Partial_Sums |.seq.|) . b₃) - 0 ) ) )
let p be real number ; :: thesis: ( p <= 0 or ex b₁ being Element of NAT st
for b₂ being Element of NAT holds
( not b₁ <= b₂ or not p <= abs (((Partial_Sums |.seq.|) . b₂) - 0 ) ) )
assume A2: 0 < p ; :: thesis: ex b₁ being Element of NAT st
for b₂ being Element of NAT holds
( not b₁ <= b₂ or not p <= abs (((Partial_Sums |.seq.|) . b₂) - 0 ) )
take 0 ; :: thesis: for b₁ being Element of NAT holds
( not 0 <= b₁ or not p <= abs (((Partial_Sums |.seq.|) . b₁) - 0 ) )
let m be Element of NAT ; :: thesis: ( not 0 <= m or not p <= abs (((Partial_Sums |.seq.|) . m) - 0 ) )
abs (((Partial_Sums |.seq.|) . m) - 0 ) = abs (0 - 0 ) by A1, Th25
.= 0 by ABSVALUE:def 1 ;
hence ( not 0 <= m or not p <= abs (((Partial_Sums |.seq.|) . m) - 0 ) ) by A2; :: thesis: verum