
theorem
  for M being non empty MetrSpace,Sm being sequence of M, St being
  sequence of TopSpaceMetr(M) st Sm=St & Sm is convergent holds lim Sm = lim St
proof
  let M be non empty MetrSpace,Sm be sequence of M, St be sequence of
  TopSpaceMetr(M);
  assume that
A1: Sm=St and
A2: Sm is convergent;
  set x=lim Sm;
  reconsider x9=x as Point of TopSpaceMetr(M) by TOPMETR:12;
  Sm is_convergent_in_metrspace_to x by A2,METRIC_6:12;
  then TopSpaceMetr(M) is T_2 & St is_convergent_to x9 by A1,Th28,PCOMPS_1:34;
  hence thesis by Th25;
end;
