let X be non empty MetrSpace; :: thesis: for x being Element of X
for S being sequence of X holds
( S is_convergent_in_metrspace_to x iff for r being Real st 0 < r holds
Ball (x,r) contains_almost_all_sequence S )
let x be Element of X; :: thesis: for S being sequence of X holds
( S is_convergent_in_metrspace_to x iff for r being Real st 0 < r holds
Ball (x,r) contains_almost_all_sequence S )
let S be sequence of X; :: thesis: ( S is_convergent_in_metrspace_to x iff for r being Real st 0 < r holds
Ball (x,r) contains_almost_all_sequence S )
thus ( S is_convergent_in_metrspace_to x implies for r being Real st 0 < r holds
Ball (x,r) contains_almost_all_sequence S ) by Th15; :: thesis: ( ( for r being Real st 0 < r holds
Ball (x,r) contains_almost_all_sequence S ) implies S is_convergent_in_metrspace_to x )
thus ( ( for r being Real st 0 < r holds
Ball (x,r) contains_almost_all_sequence S ) implies S is_convergent_in_metrspace_to x ) :: thesis: verum

proof

assume for r being Real st 0 < r holds
Ball (x,r) contains_almost_all_sequence S ; :: thesis: S is_convergent_in_metrspace_to x
then for V being Subset of X st x in V & V in Family_open_set X holds
V contains_almost_all_sequence S by Th16;
hence S is_convergent_in_metrspace_to x by Th17; :: thesis: verum

end;