
theorem Th33:
  for T being non empty TopStruct, S being sequence of T, x being
Point of T, Y being Subset of T st Y = {y where y is Point of T : x in Cl({y})
  } & rng S c= Y holds S is_convergent_to x
proof
  let T be non empty TopStruct, S be sequence of T, x be Point of T, Y be
  Subset of T;
  assume that
A1: Y = {y where y is Point of T : x in Cl({y}) } and
A2: rng S c= Y;
  let U1 be Subset of T;
  assume
A3: U1 is open & x in U1;
  take 0;
  let m be Nat;
  m in NAT by ORDINAL1:def 12;
  then m in dom S by NORMSP_1:12;
  then S.m in rng S by FUNCT_1:def 3;
  then S.m in Y by A2;
  then consider y being Point of T such that
A4: y=S.m and
A5: x in Cl({y}) by A1;
  assume 0 <= m;
  {y} meets U1 by A3,A5,PRE_TOPC:def 7;
  hence S.m in U1 by A4,ZFMISC_1:50;
end;
