let seq be Real_Sequence; :: thesis: ( seq is constant implies seq is convergent )
assume seq is constant ; :: thesis: seq is convergent
then consider r being Real such that
A1: for n being Nat holds seq . n = r by VALUED_0:def 18;
take g = r; :: according to SEQ_2:def 6 :: thesis: for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - g) < p
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 . m) - g) < p )
assume A2: 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((seq . m) - g) < p
take n = 0 ; :: thesis: for m being Element of NAT st n <= m holds
abs ((seq . m) - g) < p
let m be Element of NAT ; :: thesis: ( n <= m implies abs ((seq . m) - g) < p )
assume n <= m ; :: thesis: abs ((seq . m) - g) < p
abs ((seq . m) - g) = abs (r - g) by A1
.= 0 by ABSVALUE:2 ;
hence abs ((seq . m) - g) < p by A2; :: thesis: verum