let X be RealNormSpace; :: thesis: for S being sequence of X
for St being sequence of (MetricSpaceNorm X) st S = St & St is convergent holds
lim St = lim S
let S be sequence of X; :: thesis: for St being sequence of (MetricSpaceNorm X) st S = St & St is convergent holds
lim St = lim S
let St be sequence of (MetricSpaceNorm X); :: thesis: ( S = St & St is convergent implies lim St = lim S )
assume that
A1: S = St and
A2: St is convergent ; :: thesis: lim St = lim S
reconsider xt = lim S as Point of (MetricSpaceNorm X) ;
S is convergent by A1, A2, Th5;
then for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - (lim S)).|| < r by NORMSP_1:def 7;
then St is_convergent_in_metrspace_to xt by A1, Th4;
hence lim St = lim S by METRIC_6:11; :: thesis: verum