let r be Real; :: thesis: for seq being Real_Sequence st ( for n being Nat holds seq . n = 1 / (n + r) ) holds
lim seq = 0
let seq be Real_Sequence; :: thesis: ( ( for n being Nat holds seq . n = 1 / (n + r) ) implies lim seq = 0 )
assume A1: for n being Nat holds seq . n = 1 / (n + r) ; :: thesis: lim seq = 0
then A2: seq is convergent by Th28;
for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((seq . m) - 0).| < p by A1, LmTh28;
hence lim seq = 0 by A2, SEQ_2:def 7; :: thesis: verum