reserve c, c1, d for Real,
  k for Nat,
  n, m, N, n1, N1, N2, N3, N4, N5, M for Element of NAT,
  x for set;

theorem Th3:
  for f being eventually-nonnegative Real_Sequence st f is
  convergent holds 0 <= lim f
proof
  let f be eventually-nonnegative Real_Sequence such that
A1: f is convergent and
A2: not 0<=(lim f);
  0<-(lim f) by A2,XREAL_1:58;
  then consider N1 being Nat such that
A3: for m being Nat st N1<=m holds |.f.m-(lim f).|<-(lim f)
          by A1,SEQ_2:def 7;
  consider N being Nat such that
A4: for n being Nat st n >= N holds 0<=f.n by Def2;
  set n1 = max( N, N1 );
A0: n1 is Nat by TARSKI:1;
A5: now
    assume f.n1=0;
    then |.f.n1-(lim f).| = -(lim f) by A2,ABSVALUE:def 1;
    hence contradiction by A0,A3,XXREAL_0:25;
  end;
  |.f.n1-(lim f).|<=-(lim f) by A0,A3,XXREAL_0:25;
  then f.n1-(lim f)<=-(lim f) by ABSVALUE:5;
  then f.n1-(lim f)+(lim f)<=-(lim f)+(lim f) by XREAL_1:6;
  hence contradiction by A0, A4,A5,XXREAL_0:25;
end;
