
theorem CLOSE01:
  for X be RealNormSpace,
      x be Point of X,
      seq be sequence of X
  st for n be Nat holds seq.n = x
  holds seq is convergent & lim seq = x
  proof
    let X be RealNormSpace,
        x be Point of X,
        seq be sequence of X;
    assume
    A1: for n be Nat holds seq.n = x;
    now
      let z,w be object;
      assume z in dom seq & w in dom seq; then
      reconsider n1 = z, n2 = w as Nat;
      thus seq.z = seq.n1
                .= x by A1
                .= seq.n2 by A1
                .= seq.w;
    end; then
    A2: seq is constant by FUNCT_1:def 10;
    hence seq is convergent by NDIFF_1:18;
    thus lim seq = seq.0 by A2,NDIFF_1:18
                .= x by A1;
  end;
