reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k,m for Element of NAT;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th6:
  (for n holds x0-1/(n+1)<seq.n & seq.n<x0 & seq.n in dom f)
  implies seq is convergent & lim seq=x0 & rng seq c= dom f \ {x0}
proof
  assume
A1: for n holds x0-1/(n+1)<seq.n & seq.n<x0 & seq.n in dom f;
A2: for n being Nat holds x0-1/(n+1)<seq.n & seq.n<x0 & seq.n in dom f
    proof let n be Nat;
      n in NAT by ORDINAL1:def 12;
     hence thesis by A1;
    end;
  hence seq is convergent & lim seq=x0 by LIMFUNC2:5;
  rng seq c=dom f/\left_open_halfline(x0) by LIMFUNC2:5,A2;
  hence thesis by Th1;
end;
