Show that C0 is a closed subset of l infinitive.
C0 is the subset of l infinitive consisting of all sequences that converge to 0.
l infinitive is the collection of all bounded real sequences.
Hint: If x(n) is a sequence in c0 converging to x belongs to l infinitive, note that
|xk| <= |xk-xk(n)| + |xk(n)| and now choose n so that |xk-xk(n)| is small independent of k.
Thanks.