let X be RealUnitarySpace; :: thesis: for g being Point of X
for seq being sequence of X st seq is convergent & lim seq = g holds
dist seq,g is convergent
let g be Point of X; :: thesis: for seq being sequence of X st seq is convergent & lim seq = g holds
dist seq,g is convergent
let seq be sequence of X; :: thesis: ( seq is convergent & lim seq = g implies dist seq,g is convergent )
assume A1: ( seq is convergent & lim seq = g ) ; :: thesis: dist seq,g is convergent

now

let r be real number ; :: thesis: ( r > 0 implies ex k being Element of NAT st
for n being Element of NAT st k <= n holds
abs (((dist seq,g) . n) - 0 ) < r )
assume A2: r > 0 ; :: thesis: ex k being Element of NAT st
for n being Element of NAT st k <= n holds
abs (((dist seq,g) . n) - 0 ) < r
r is Real by XREAL_0:def 1;
then consider m1 being Element of NAT such that
A3: for n being Element of NAT st n >= m1 holds
dist (seq . n),g < r by A1, A2, Def2;
take k = m1; :: thesis: for n being Element of NAT st k <= n holds
abs (((dist seq,g) . n) - 0 ) < r

now

let n be Element of NAT ; :: thesis: ( n >= k implies abs (((dist seq,g) . n) - 0 ) < r )
dist (seq . n),g >= 0 by BHSP_1:44;
then A4: abs ((dist (seq . n),g) - 0 ) = dist (seq . n),g by ABSVALUE:def 1;
assume n >= k ; :: thesis: abs (((dist seq,g) . n) - 0 ) < r
then dist (seq . n),g < r by A3;
hence abs (((dist seq,g) . n) - 0 ) < r by A4, Def4; :: thesis: verum

end;

hence for n being Element of NAT st k <= n holds
abs (((dist seq,g) . n) - 0 ) < r ; :: thesis: verum

end;

hence dist seq,g is convergent by SEQ_2:def 6; :: thesis: verum