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;

theorem Th2:
  seq is non-zero & seq is convergent & lim seq=0 & seq is
  non-decreasing implies for n holds seq.n<0
proof
  assume that
A1: seq is non-zero and
A2: seq is convergent & lim seq=0 and
A3: seq is non-decreasing and
A4: ex n st not seq.n<0;
  consider n such that
A5: not seq.n<0 by A4;
  now
    per cases by A5;
    suppose
A6:   0<seq.n;
      then consider n1 be Nat such that
A7:   for m st n1<=m holds |.seq.m-0.|<seq.n by A2,SEQ_2:def 7;
      |.seq.(n1+n)-0.|<seq.n by A7,NAT_1:12;
      then n<=n1+n & seq.(n1+n)<seq.n by A6,ABSVALUE:def 1,NAT_1:12;
      hence contradiction by A3,SEQM_3:6;
    end;
    suppose
      seq.n=0;
      hence contradiction by A1,SEQ_1:5;
    end;
  end;
  hence contradiction;
end;
