 reserve X for RealUnitarySpace;
 reserve x, y, y1, y2 for Point of X;

theorem Th11:
for M being RealUnitarySpace
for X being Subset of TopSpaceNorm RUSp2RNSp M holds
( X is closed iff
for S being sequence of M
st ( for n being Nat holds S . n in X ) & S is convergent holds
lim S in X )
proof
let M be RealUnitarySpace;
let X be Subset of TopSpaceNorm RUSp2RNSp M;
hereby assume A1p: X is closed;
thus for S being sequence of M
st ( for n being Nat holds S . n in X )
& S is convergent holds lim S in X
proof
let S be sequence of M;
 assume that A2: for n being Nat holds S . n in X and
   A3: S is convergent;
reconsider St = S as sequence of MetricSpaceNorm RUSp2RNSp M;
 A5: St is convergent by A3,Th5; then
 lim St = lim S by Th6;
 hence lim S in X by A2,A5,A1p,TOPMETR4:6;
end;
end;
assume A6:for S being sequence of M
st ( for n being Nat holds S . n in X ) & S is convergent holds lim S in X;
for St being sequence of MetricSpaceNorm RUSp2RNSp M
st ( for n being Nat holds St . n in X )
& St is convergent holds lim St in X
proof
  let St be sequence of MetricSpaceNorm RUSp2RNSp M;
  assume that
  A7: for n being Nat holds St . n in X and
  A8: St is convergent;
  reconsider S = St as sequence of M;
  A9: for n being Nat holds S . n in X by A7;
  A10: S is convergent by A8,Th5;
  lim St = lim S by A8,Th6;
  hence lim St in X by A6,A9,A10;
end;
hence X is closed by TOPMETR4:6;
end;
