reserve n for Nat;

theorem Th25:
  for S being SetSequence of the carrier of TOP-REAL n, P being
Subset of TOP-REAL n st P is bounded & (for i being Nat holds S.i c=
  P) holds Lim_inf S is bounded
proof
  let S be SetSequence of the carrier of TOP-REAL n;
  let P be Subset of TOP-REAL n;
  assume that
A1: P is bounded and
A2: for i being Nat holds S.i c= P;
  reconsider P9= P as bounded Subset of Euclid n by A1,JORDAN2C:11;
  consider t being Real, z being Point of Euclid n such that
A3: 0 < t and
A4: P9 c= Ball (z,t) by METRIC_6:def 3;
  set r = 1, R = r + r + 3*t;
  assume Lim_inf S is non bounded;
  then consider x, y being Point of Euclid n such that
A5: x in Lim_inf S and
A6: y in Lim_inf S and
A7: dist (x, y) > R by A3,Th8;
  consider k1 being Nat such that
A8: for m being Nat st m > k1 holds S.m meets Ball (x, r) by A5,Th14;
  consider k2 being Nat such that
A9: for m being Nat st m > k2 holds S.m meets Ball (y, r) by A6,Th14;
  set k = max (k1, k2) + 1;
  S.k c= P9 by A2;
  then
A10: S.k c= Ball (z,t) by A4;
  2*t < 3*t by A3,XREAL_1:68;
  then
A11: r + r + 2*t < r + r + 3*t by XREAL_1:8;
  max (k1,k2) >= k2 by XXREAL_0:25;
  then k > k2 by NAT_1:13;
  then
A12: Ball (z,t) meets Ball (y, r) by A9,A10,XBOOLE_1:63;
  max (k1,k2) >= k1 by XXREAL_0:25;
  then k > k1 by NAT_1:13;
  then Ball (z,t) meets Ball (x, r) by A8,A10,XBOOLE_1:63;
  hence thesis by A7,A12,A11,Th10,XXREAL_0:2;
end;
