let seq be Real_Sequence; :: thesis: ( seq is non-increasing & not seq is bounded_below implies seq is divergent_to-infty )
assume that
A1: seq is non-increasing and
A2: not seq is bounded_below ; :: thesis: seq is divergent_to-infty
let r be Real; :: according to LIMFUNC1:def 5 :: thesis: ex n being Nat st
for m being Nat st n <= m holds
seq . m < r
consider n being Nat such that
A3: seq . n <= r - 1 by A2;
take n ; :: thesis: for m being Nat st n <= m holds
seq . m < r
let m be Nat; :: thesis: ( n <= m implies seq . m < r )
assume n <= m ; :: thesis: seq . m < r
then seq . m <= seq . n by A1, SEQM_3:8;
then seq . m <= r - 1 by A3, XXREAL_0:2;
hence seq . m < r by Lm1; :: thesis: verum