reserve n,m,k,i for Nat,
  g,s,t,p for Real,
  x,y,z for object, X,Y,Z for set,
  A1 for SetSequence of X,
  F1 for FinSequence of bool X,
  RFin for real-valued FinSequence,
  Si for SigmaField of X,
  XSeq,YSeq for SetSequence of Si,
  Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq,BSeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th3:
  for f being Real_Sequence st (ex k st for n st k<=n holds f.n=g)
  holds f is convergent & lim f = g
proof
  let f be Real_Sequence;
  given k such that
A1: for n st k<=n holds f.n=g;
A2: now
    let p such that
A3: 0 < p;
    take k;
      let m be Nat;
      assume k<=m;
      then f.m = g by A1;
      hence |.f.m-g.|<p by A3,ABSVALUE:2;
  end;
  hence f is convergent by SEQ_2:def 6;
  hence thesis by A2,SEQ_2:def 7;
end;
