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 Th3:
  seq is non-zero & seq is convergent & lim seq=0 & seq is
  non-increasing implies for n holds 0<seq.n
proof
  assume that
A1: seq is non-zero and
A2: seq is convergent & lim seq=0 and
A3: seq is non-increasing and
A4: ex n st not 0<seq.n;
  consider n such that
A5: not 0<seq.n by A4;
  now
    per cases by A5;
    suppose
A6:   seq.n<0;
      then -0<-seq.n by XREAL_1:24;
      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;
A8:   |.seq.(n1+n)-0.|<-seq.n by A7,NAT_1:12;
A9:   n<=n1+n by NAT_1:12;
      then seq.(n1+n)<0 by A3,A6,SEQM_3:8;
      then -seq.(n1+n)<-seq.n by A8,ABSVALUE:def 1;
      then seq.n<seq.(n1+n) by XREAL_1:24;
      hence contradiction by A3,A9,SEQM_3:8;
    end;
    suppose
      seq.n=0;
      hence contradiction by A1,SEQ_1:5;
    end;
  end;
  hence contradiction;
end;
