reserve n for Nat;

theorem
  for n being Nat, r being Point of TOP-REAL n, X being
Subset of TOP-REAL n st r in Cl X holds ex seq being Real_Sequence of n st rng
  seq c= X & seq is convergent & lim seq = r
proof
  let n be Nat, r be Point of TOP-REAL n, X be Subset of TOP-REAL n;
  reconsider r9 = r as Point of Euclid n by TOPREAL3:8;
  defpred P[object,object] means ex z being Nat st $1 = z &
   $2 = the Element of X /\ Ball (r9, 1/(z+1));
  assume
A1: r in Cl X;
A2: now
    let x be object;
    assume x in NAT;
    then reconsider k = x as Nat;
    set n1 = k+1;
    set oi = Ball (r9, 1/n1);
    reconsider oi as open Subset of TOP-REAL n by Th1;
    reconsider u = the Element of X /\ oi as object;
    take u;
    dist (r9,r9) < 1/n1 by METRIC_1:1;
    then r in oi by METRIC_1:11;
    then X meets oi by A1,PRE_TOPC:24;
    then X /\ oi is non empty;
    then u in X /\ oi;
    hence u in the carrier of TOP-REAL n;
    thus P[x,u];
  end;
  consider seq being Function such that
A3: dom seq = NAT & rng seq c= the carrier of TOP-REAL n and
A4: for x being object st x in NAT holds P[x,seq.x] from FUNCT_1:sch 6(A2);
  reconsider seq as Real_Sequence of n by A3,FUNCT_2:def 1,RELSET_1:4;
  take seq;
  thus rng seq c= X
  proof
    let y be object;
    assume y in rng seq;
    then consider x being object such that
A5: x in dom seq and
A6: seq.x = y by FUNCT_1:def 3;
    consider k being Nat such that
    x = k and
A7: seq.x = the Element of X /\ Ball (r9,1/(k+1)) by A4,A5;
    set n1 = k+1;
    reconsider oi = Ball (r9,1/n1) as open Subset of TOP-REAL n by Th1;
    dist (r9,r9) < 1/n1 by METRIC_1:1;
    then r in oi by METRIC_1:11;
    then X meets oi by A1,PRE_TOPC:24;
    then X /\ oi is non empty;
    hence thesis by A6,A7,XBOOLE_0:def 4;
  end;
A8: now
    let p be Real;
    set cp = [/ 1/p \];
A9: 1/p <= cp by INT_1:def 7;
    assume
A10: 0 < p;
    then
A11: 0 < cp by INT_1:def 7;
    then reconsider cp as Element of NAT by INT_1:3;
     reconsider cp as Nat;
    take k = cp;
    k < k+1 by NAT_1:13;
    then
A12: 1/(k+1) < 1/k by A11,XREAL_1:88;
    1/(1/p) >= 1/cp by A10,A9,XREAL_1:85;
    then
A13: 1/(k+1) < p by A12,XXREAL_0:2;
    let m be Nat;
    assume k <= m;
    then
A14: k+1 <= m+1 by XREAL_1:6;
    set m1 = m+1;
    1/m1 <= 1/(k+1) by A14,XREAL_1:85;
    then
A15: 1/m1 < p by A13,XXREAL_0:2;
    set oi = Ball (r9,1/m1);
    reconsider oi as open Subset of TOP-REAL n by Th1;
    dist (r9,r9) < 1/m1 by METRIC_1:1;
    then r in oi by METRIC_1:11;
    then X meets oi by A1,PRE_TOPC:24;
    then
A16: X /\ oi is non empty;
    m in NAT by ORDINAL1:def 12;
    then
    ex m9 being Nat st m9 = m & seq.m = the Element of X /\ Ball (r9,1
    /(m9+1)) by A4;
    then seq.m in oi by A16,XBOOLE_0:def 4;
    hence |. seq.m - r .| < p by A15,Th2,XXREAL_0:2;
  end;
  hence seq is convergent by TOPRNS_1:def 8;
  hence thesis by A8,TOPRNS_1:def 9;
end;
