reserve n for Nat;

theorem
  for S being SetSequence of the carrier of TOP-REAL n, p being Point of
  TOP-REAL n holds (ex s being Real_Sequence of n st s is convergent & (for x
  being Nat holds s.x in S.x) & p = lim s) implies p in Lim_inf S
proof
  let S be SetSequence of the carrier of TOP-REAL n, p be Point of TOP-REAL n;
  reconsider p9 = p as Point of Euclid n by TOPREAL3:8;
  given s being Real_Sequence of n such that
A1: s is convergent and
A2: for x being Nat holds s.x in S.x and
A3: p = lim s;
  for r being Real st r > 0 ex k being Nat st for m
  being Nat st m > k holds S.m meets Ball (p9, r)
  proof
    let r be Real;
    reconsider r9 = r as Real;
    assume r > 0;
    then consider l being Nat such that
A4: for m being Nat st l <= m holds |. s.m - p .| < r9 by A1,A3,
TOPRNS_1:def 9;
    reconsider v = max (0, l) as Nat by TARSKI:1;
    take v;
    let m be Nat;
    assume v < m;
    then l <= m by XXREAL_0:30;
    then |. s.m - p .| < r9 by A4;
    then
A5: s.m in Ball (p9, r) by Th3;
    s.m in S.m by A2;
    hence thesis by A5,XBOOLE_0:3;
  end;
  hence thesis by Th14;
end;
