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 Th4:
  seq is convergent & 0<lim seq implies ex n st for m st n<=m holds 0<seq.m
proof
  assume that
A1: seq is convergent and
A2: 0<lim seq and
A3: for n ex m st n<=m & not 0<seq.m;
  consider n such that
A4: for m st n<=m holds |.seq.m-lim seq.|<lim seq by A1,A2,SEQ_2:def 7;
  consider m such that
A5: n<=m and
A6: not 0<seq.m by A3;
A7: |.seq.m-lim seq.|<lim seq by A4,A5;
  now
    per cases by A6;
    suppose
A8:   seq.m<0;
      then -(seq.m-lim seq)<lim seq by A2,A7,ABSVALUE:def 1;
      then lim seq-seq.m<lim seq;
      then lim seq<lim seq+seq.m by XREAL_1:19;
      then lim seq - lim seq<seq.m by XREAL_1:19;
      hence contradiction by A8;
    end;
    suppose
      seq.m=0;
      then |.-lim seq.|<lim seq by A7;
      then |.lim seq.|<lim seq by COMPLEX1:52;
      hence contradiction by A2,ABSVALUE:def 1;
    end;
  end;
  hence contradiction;
end;
