 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 Th21:
for X being NormedLinearTopSpace,
    S being sequence of X,
    x being Point of X holds
 S is convergent & x=lim S
iff
for r being Real st 0 < r holds
  ex m being Nat st
  for n being Nat st m <= n holds
    ||.((S . n) - x).|| < r
proof
let X be NormedLinearTopSpace,
    S be sequence of X,
    x be Point of X;
hereby assume
  S is convergent & lim S = x; then
  S is_convergent_to x by FRECHET2:25;
  hence for r being Real st 0 < r holds
  ex m being Nat st
  for n being Nat st m <= n holds
    ||.((S . n) - x).|| < r by Th20;
end;
assume for r being Real st 0 < r holds
  ex m being Nat st
  for n being Nat st m <= n holds
    ||.((S . n) - x).|| < r;
then S is_convergent_to x by Th20;
hence S is convergent & x=lim S by FRECHET2:25;
end;
