 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th22:
for X being NormedLinearTopSpace,
    S being sequence of X
 st S is convergent
holds
  for r being Real st 0 < r holds
  ex m being Nat st
  for n being Nat st m <= n holds
    ||.(S . n) - (lim S) .|| < r
proof
let X be NormedLinearTopSpace,
    S be sequence of X;
assume S is convergent;
then consider x being Point of X such that
  A1: S is_convergent_to x;
S is convergent & x = lim S by FRECHET2:25,A1;
hence thesis by A1,Th20;
end;
