let X be RealNormSpace; :: thesis: for S being sequence of X
for St being sequence of (LinearTopSpaceNorm X)
for x being Point of X
for xt being Point of (LinearTopSpaceNorm 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 (LinearTopSpaceNorm X)
for x being Point of X
for xt being Point of (LinearTopSpaceNorm 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 (LinearTopSpaceNorm X); :: thesis: for x being Point of X
for xt being Point of (LinearTopSpaceNorm 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 (LinearTopSpaceNorm 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 (LinearTopSpaceNorm 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 ) )
reconsider xxt = xt as Point of (TopSpaceNorm X) by Def4;
assume A1: ( S = St & 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 )
the carrier of (LinearTopSpaceNorm X) = the carrier of (TopSpaceNorm X) by Def4;
then reconsider SSt = St as sequence of (TopSpaceNorm X) ;
( St is_convergent_to xt iff SSt is_convergent_to xxt ) by Th26;
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, Th12; :: thesis: verum