reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;
reserve X for non empty MetrSpace,
  x,y for Element of X,
  V for Subset of X,
  S,S1,T for sequence of X,
  Nseq for increasing sequence of NAT;

theorem
  S is convergent implies S is bounded
proof
  assume S is convergent;
  then consider x such that
A1: S is_convergent_in_metrspace_to x and
  lim S = x by Th13;
  dist_to_point(S,x) is convergent by A1,Th14;
  then dist_to_point(S,x) is bounded by SEQ_2:13;
  then consider r be Real such that
A2: 0 < r and
A3: for n being Nat holds |.dist_to_point(S,x).n.| < r by SEQ_2:3;
  reconsider r as Real;
  for n holds S.n in Ball(x,r)
  proof
    let n;
A4: dist_to_point(S,x).n = dist(S.n,x) by Def6;
    then 0 <= dist_to_point(S,x).n by METRIC_1:5;
    then |.dist_to_point(S,x).n.| = dist_to_point(S,x).n by ABSVALUE:def 1;
    then dist(S.n,x) < r by A3,A4;
    hence thesis by METRIC_1:11;
  end;
  hence thesis by A2,Th8;
end;
