
theorem
  for X be RealNormSpace, x be Point of X holds {x} is closed
  proof
    let X be RealNormSpace, x be Point of X;
    for s1 be sequence of X st rng s1 c= {x} & s1 is convergent
    holds lim s1 in {x}
    proof
      let s1 be sequence of X;
      assume
      A1: rng s1 c= {x} & s1 is convergent;
      now
        let n be Nat;
        s1.n in rng s1 by FUNCT_2:4,ORDINAL1:def 12;
        hence s1.n = x by A1,TARSKI:def 1;
      end; then
      lim s1 = x by CLOSE01;
      hence thesis by TARSKI:def 1;
    end;
    hence thesis;
  end;
