let X be non empty MetrSpace; :: thesis: for x being Element of X
for S being sequence of X st ( for V being Subset of X st x in V & V in Family_open_set X holds
V contains_almost_all_sequence S ) holds
S is_convergent_in_metrspace_to x
let x be Element of X; :: thesis: for S being sequence of X st ( for V being Subset of X st x in V & V in Family_open_set X holds
V contains_almost_all_sequence S ) holds
S is_convergent_in_metrspace_to x
let S be sequence of X; :: thesis: ( ( for V being Subset of X st x in V & V in Family_open_set X holds
V contains_almost_all_sequence S ) implies S is_convergent_in_metrspace_to x )
A1: for r being Real st 0 < r holds
x in Ball (x,r)

proof

let r be Real; :: thesis: ( 0 < r implies x in Ball (x,r) )
assume 0 < r ; :: thesis: x in Ball (x,r)
then dist (x,x) < r by METRIC_1:1;
hence x in Ball (x,r) by METRIC_1:11; :: thesis: verum

end;

assume A2: for V being Subset of X st x in V & V in Family_open_set X holds
V contains_almost_all_sequence S ; :: thesis: S is_convergent_in_metrspace_to x
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
dist ((S . n),x) < r

proof

let r be Real; :: thesis: ( 0 < r implies ex m being Nat st
for n being Nat st m <= n holds
dist ((S . n),x) < r )
assume 0 < r ; :: thesis: ex m being Nat st
for n being Nat st m <= n holds
dist ((S . n),x) < r
then x in Ball (x,r) by A1;
then Ball (x,r) contains_almost_all_sequence S by A2, PCOMPS_1:29;
then consider m1 being Nat such that
A3: for n being Nat st m1 <= n holds
S . n in Ball (x,r) ;
take k = m1; :: thesis: for n being Nat st k <= n holds
dist ((S . n),x) < r
let n be Nat; :: thesis: ( k <= n implies dist ((S . n),x) < r )
assume k <= n ; :: thesis: dist ((S . n),x) < r
then S . n in Ball (x,r) by A3;
hence dist ((S . n),x) < r by METRIC_1:11; :: thesis: verum

end;

hence S is_convergent_in_metrspace_to x ; :: thesis: verum