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.