let X be RealNormSpace; :: thesis: for S being sequence of X
for St being sequence of ()
for x being Point of X
for xt being Point of () st S = St & x = xt holds
( St is_convergent_to xt 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 )

let S be sequence of X; :: thesis: for St being sequence of ()
for x being Point of X
for xt being Point of () st S = St & x = xt holds
( St is_convergent_to xt 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 )

let St be sequence of (); :: thesis: for x being Point of X
for xt being Point of () st S = St & x = xt holds
( St is_convergent_to xt 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 )

let x be Point of X; :: thesis: for xt being Point of () st S = St & x = xt holds
( St is_convergent_to xt 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 )

let xt be Point of (); :: thesis: ( S = St & x = xt implies ( St is_convergent_to xt 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 ) )

assume that
A1: S = St and
A2: x = xt ; :: thesis: ( St is_convergent_to xt 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 )

reconsider Sm = St as sequence of () ;
reconsider xm = x as Point of () ;
( St is_convergent_to xt iff Sm is_convergent_in_metrspace_to xm ) by ;
hence ( St is_convergent_to xt 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 ) by ; :: thesis: verum