Give a counterexample to show the converse

Question

Problems:

A counterexample

1. (a) Prove that if lim_n→∞ a_n = ∞ then lim_n→∞ 1/a_n = 0
    (b) Give a counterexample to show that the converse (if lim_n→∞ a_n = 0 then lim_n→∞ 1/a_n = ∞) is false.

2. Give an example of a sequence {an} satisfying all of the following:

{a_n} is monotonic

0 < a_n < 1 for all n and no two terms are equal

lim_n→∞ a_{n =} 1/2_.