
theorem Th1:
  for X being non empty MetrSpace, S being sequence of X, F being
  Subset of TopSpaceMetr(X) st S is convergent & (for n being Nat
  holds S.n in F) & F is closed holds lim S in F
proof
  let X be non empty MetrSpace, S be sequence of X, F be Subset of
  TopSpaceMetr(X);
  assume that
A1: S is convergent and
A2: for n being Nat holds S.n in F and
A3: F is closed;
A4: for G being Subset of TopSpaceMetr(X) st G is open holds lim S in G
  implies F meets G
  proof
    let G be Subset of TopSpaceMetr(X);
    assume
A5: G is open;
    now
      assume lim S in G;
      then consider r1 being Real such that
A6:   r1>0 and
A7:   Ball(lim S,r1) c= G by A5,TOPMETR:15;
      reconsider r2=r1 as Real;
      consider n being Nat such that
A8:   for m being Nat st m>=n holds dist(S.m,lim S)<r2 by A1,A6,
TBSP_1:def 3;
      dist(S.n,lim S)<r2 by A8;
      then
A9:   S.n in Ball(lim S,r1) by METRIC_1:11;
      S.n in F by A2;
      then S.n in F /\ G by A7,A9,XBOOLE_0:def 4;
      hence F meets G by XBOOLE_0:def 7;
    end;
    hence thesis;
  end;
  reconsider F0=F as Subset of TopSpaceMetr(X);
  lim S in the carrier of X;
  then lim S in the carrier of TopSpaceMetr(X) by TOPMETR:12;
  then lim S in Cl F0 by A4,PRE_TOPC:def 7;
  hence thesis by A3,PRE_TOPC:22;
end;
