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 Th2:
  (for n holds 0<|.x0-seq.n.| & |.x0-seq.n.|<1/(n+1) & seq.n in
  dom f) implies seq is convergent & lim seq=x0 & rng seq c=dom f & rng seq c=
  dom f \ {x0}
proof
  assume
A1: for n holds 0<|.x0-seq.n.| & |.x0-seq.n.|<1/(n+1) & seq.n in dom f;
A2: now
    let r be Real such that
A3: 0<r;
    consider n being Nat such that
A4: r"<n by SEQ_4:3;
    take n;
    let k be Nat;
    assume n<=k;
    then n+1<=k+1 by XREAL_1:6;
    then
A5: 1/(k+1)<=1/(n+1) by XREAL_1:118;
    n<=n+1 by NAT_1:12;
    then r"<n+1 by A4,XXREAL_0:2;
    then 1/(n+1)<1/r" by A3,XREAL_1:76;
    then 1/(k+1)<1/r" by A5,XXREAL_0:2;
    then
A6: 1/(k+1)<r by XCMPLX_1:216;
    k in NAT by ORDINAL1:def 12;
    then |.x0-seq.k.|<1/(k+1) by A1;
    then |.-(seq.k-x0).|<r by A6,XXREAL_0:2;
    hence |.seq.k-x0.|<r by COMPLEX1:52;
  end;
  hence seq is convergent by SEQ_2:def 6;
  hence lim seq=x0 by A2,SEQ_2:def 7;
  thus
A7: rng seq c=dom f
  proof
    let x be object;
    assume x in rng seq;
    then ex n st seq.n=x by FUNCT_2:113;
    hence thesis by A1;
  end;
  let x be object;
  assume
A8: x in rng seq;
  then consider n such that
A9: seq.n=x by FUNCT_2:113;
  0<>|.x0-seq.n.| by A1;
  then x0-seq.n+seq.n<>0+seq.n by ABSVALUE:2;
  then not x in {x0} by A9,TARSKI:def 1;
  hence thesis by A7,A8,XBOOLE_0:def 5;
end;
