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 & lim (dist (seq,g)) = 0 )
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 & lim (dist (seq,g)) = 0 )
let seq be sequence of X; :: thesis: ( seq is convergent & lim seq = g implies ( dist (seq,g) is convergent & lim (dist (seq,g)) = 0 ) )
assume A1: ( seq is convergent & lim seq = g ) ; :: thesis: ( dist (seq,g) is convergent & lim (dist (seq,g)) = 0 )

A2: now

let r be real number ; :: thesis: ( r > 0 implies ex k being Element of NAT st
for n being Element of NAT st n >= k holds
abs (((dist (seq,g)) . n) - 0) < r )
assume A3: r > 0 ; :: thesis: ex k being Element of NAT st
for n being Element of NAT st n >= k 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
A4: for n being Element of NAT st n >= m1 holds
dist ((seq . n),g) < r by A1, A3, Def2;
take k = m1; :: thesis: for n being Element of NAT st n >= k holds
abs (((dist (seq,g)) . n) - 0) < r
let n be Element of NAT ; :: thesis: ( n >= k implies abs (((dist (seq,g)) . n) - 0) < r )
dist ((seq . n),g) >= 0 by BHSP_1:37;
then A5: 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 A4;
hence abs (((dist (seq,g)) . n) - 0) < r by A5, Def4; :: thesis: verum

end;

dist (seq,g) is convergent by A1, Th23;
hence ( dist (seq,g) is convergent & lim (dist (seq,g)) = 0 ) by A2, SEQ_2:def 7; :: thesis: verum