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

theorem Th10:
for M being RealUnitarySpace
for X being Subset of TopSpaceNorm RUSp2RNSp M holds
for x being object holds
( x in Cl X iff ex S being sequence of M st
( ( for n being Nat holds S . n in X ) & S is convergent & lim S = x ) )
proof
let M be RealUnitarySpace;
let X be Subset of TopSpaceNorm RUSp2RNSp M;
let x be object;
hereby assume x in Cl X; then
consider St being sequence of MetricSpaceNorm RUSp2RNSp M
such that
 A1: for n being Nat holds St . n in X and
 A2: St is convergent & lim St = x by TOPMETR4:5;
reconsider S = St as sequence of M;
take S;
 thus for n being Nat holds S . n in X by A1;
 thus S is convergent by A2,Th5;
 thus lim S = x by Th6,A2;
end;
assume
ex S being sequence of M st
( ( for n being Nat holds S . n in X )
& S is convergent & lim S = x ); then
consider S being sequence of M such that
A3:( for n being Nat holds S . n in X )
    & S is convergent & lim S = x;
reconsider St = S as sequence of MetricSpaceNorm RUSp2RNSp M;
A5: St is convergent by A3,Th5;
 then lim St = x by A3,Th6;
  hence x in Cl X by TOPMETR4:5,A3,A5;
end;
