 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 Th20:
for X being NormedLinearTopSpace,
    S being sequence of X,
    x being Point of X holds
 S is_convergent_to x
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;
consider RNS be RealNormSpace such that
A1:  RNS = the NORMSTR of X
  & the topology of X = the topology of (TopSpaceNorm RNS) by Def7;
reconsider N = S as sequence of RNS by A1;
reconsider xn = x as Point of RNS by A1;
reconsider T = S as sequence of TopSpaceNorm RNS by A1;
reconsider xt = x as Point of TopSpaceNorm RNS by A1;
A2:  S is_convergent_to x
  iff
   T is_convergent_to xt
proof
hereby assume A3: S is_convergent_to x;
now
  let U1 be Subset of TopSpaceNorm RNS;
  assume A4: U1 is open & xt in U1;
  reconsider U0 = U1 as Subset of X by A1;
  U0 is open & x in U0 by A1,A4;
  then consider n being Nat such that
  A5: for m being Nat st n <= m holds S . m in U0 by A3;
  take n;
  let m be Nat;
  assume n <= m;
  hence S . m in U1 by A5;
 end;
hence T is_convergent_to xt;
end;
assume
A6:T is_convergent_to xt;
now
let U1 be Subset of X;
   assume A7:U1 is open & x in U1;
 reconsider U0 = U1 as Subset of TopSpaceNorm RNS by A1;
  U0 is open & xt in U0 by A1,A7; then
  consider n being Nat such that
  A8: for m being Nat st n <= m holds T . m in U0 by A6;
  take n;
  let m be Nat;
  assume n <= m;
  hence S . m in U1 by A8;
end;
hence S is_convergent_to x;
end;
(for r being Real st 0 < r holds
  ex m being Nat st
  for n being Nat st m <= n holds
    ||.((N . n) - xn).|| < r )
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
hereby assume
A9: for r being Real st 0 < r holds
  ex m being Nat st
  for n being Nat st m <= n holds ||.((N . n) - xn).|| < r;
  let r be Real;
  assume 0 < r; then
  consider m being Nat such that
  A10:   for n being Nat st m <= n holds ||.((N . n) - xn).|| < r by A9;
  take m;
   let n be Nat;
   assume m <= n; then
   ||.((N . n) - xn).|| < r by A10;
   hence ||.((S . n) - x).|| < r by Th19,A1;
   end;
assume
A12: 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;
  let r be Real;
  assume 0 < r; then
  consider m being Nat such that
  A13:   for n being Nat st m <= n holds ||.((S . n) - x).|| < r by A12;
  take m;
   let n be Nat;
     assume m <= n; then
   ||.((S . n) - x).|| < r by A13;
   hence ||.((N . n) - xn).|| < r by Th19,A1;
 end;
hence thesis by A2,NORMSP_2:12;
end;
