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 Th10:
  S is convergent implies ex x st S is_convergent_in_metrspace_to x
proof
  assume S is convergent;
  then consider x such that
A1: for r st 0 < r ex m st for n st m <= n holds dist(S.n,x) < r;
  S is_convergent_in_metrspace_to x by A1;
  hence thesis;
end;
