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 Th11:
  S is_convergent_in_metrspace_to x implies lim S = x
proof
  assume S is_convergent_in_metrspace_to x; then
  S is convergent & for r st 0 < r ex m st for n st m <= n holds dist(S.n,
  x) < r;
  hence thesis by TBSP_1:def 3;
end;
