let X be RealNormSpace; :: thesis: for S being sequence of X

for St being sequence of (TopSpaceNorm X)

for x being Point of X

for xt being Point of (TopSpaceNorm X) 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 (TopSpaceNorm X)

for x being Point of X

for xt being Point of (TopSpaceNorm X) 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 (TopSpaceNorm X); :: thesis: for x being Point of X

for xt being Point of (TopSpaceNorm X) 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 (TopSpaceNorm X) 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 (TopSpaceNorm X); :: 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 (MetricSpaceNorm X) ;

reconsider xm = x as Point of (MetricSpaceNorm X) ;

( St is_convergent_to xt iff Sm is_convergent_in_metrspace_to xm ) by A2, FRECHET2:28;

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 A1, Th4; :: thesis: verum

for St being sequence of (TopSpaceNorm X)

for x being Point of X

for xt being Point of (TopSpaceNorm X) 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 (TopSpaceNorm X)

for x being Point of X

for xt being Point of (TopSpaceNorm X) 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 (TopSpaceNorm X); :: thesis: for x being Point of X

for xt being Point of (TopSpaceNorm X) 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 (TopSpaceNorm X) 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 (TopSpaceNorm X); :: 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 (MetricSpaceNorm X) ;

reconsider xm = x as Point of (MetricSpaceNorm X) ;

( St is_convergent_to xt iff Sm is_convergent_in_metrspace_to xm ) by A2, FRECHET2:28;

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 A1, Th4; :: thesis: verum