reserve Omega for set;
reserve X, Y, Z, p,x,y,z for set;
reserve D, E for Subset of Omega;
reserve f for Function;
reserve m,n for Nat;
reserve r,r1 for Real;
reserve seq for Real_Sequence;

theorem Th1:
  for r,seq st (ex n st for m st n <= m holds seq.m = r) holds seq
  is convergent & lim seq = r
proof
  let r,seq such that
A1: ex n st for m st n <= m holds seq.m = r;
A2: for r1 st 0 < r1 ex n st for m st n <= m holds |.seq.m-r.|<r1
  proof
    consider n such that
A3: for m st n <= m holds seq.m = r by A1;
    let r1 such that
A4: 0 < r1;
    take n;
    let m;
    assume n <= m;
    then seq.m = r by A3;
    hence thesis by A4,ABSVALUE:2;
  end;
  then seq is convergent by SEQ_2:def 6;
  hence thesis by A2,SEQ_2:def 7;
end;
