reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem Th36:
  seq is convergent & lim seq=0 & (ex k st for n st k<=n holds seq
  .n<0) implies seq" is divergent_to-infty
proof
  assume
A1: seq is convergent & lim seq=0;
  given k such that
A2: for n st k<=n holds seq.n<0;
  let r;
  set l=|.r.|+1;
  0<=|.r.| by COMPLEX1:46;
  then consider o be Nat such that
A3: for n st o<=n holds |.seq.n-0.|<l" by A1,SEQ_2:def 7;
  reconsider m=max(k,o) as Nat by TARSKI:1;
  take m;
  let n;
  assume
A4: m<=n;
  k<=m by XXREAL_0:25;
  then k<= n by A4,XXREAL_0:2;
  then
A5: seq.n<0 by A2;
  then
A6: 0<-(seq.n) by XREAL_1:58;
  o<=m by XXREAL_0:25;
  then o<=n by A4,XXREAL_0:2;
  then |.seq.n-0.|<l" by A3;
  then -(seq.n)<l" by A5,ABSVALUE:def 1;
  then 1/(l")<1/(-(seq.n)) by A6,XREAL_1:76;
  then l<1/(-(seq.n)) by XCMPLX_1:216;
  then l<(-(seq.n))" by XCMPLX_1:215;
  then l<-((seq.n)") by XCMPLX_1:222;
  then
A7: --((seq.n)")<-l by XREAL_1:24;
  -|.r.|<=r by ABSVALUE:4;
  then -|.r.|-1<r by Lm1;
  then (seq.n)"<r by A7,XXREAL_0:2;
  hence thesis by VALUED_1:10;
end;
