Prove the converse part of Theorem.Let (sn) be a sequence of positive numbers. Then lim sn...

Prove the converse part of Theorem.

Let (sn) be a sequence of positive numbers. Then lim sn = +∞ if and only if lim (1/sn) = 0.

Proof: Suppose that lim sn = +∞. Given any ε > 0, let M = 1 /ε. Then there exists a natural number N such that n ≥ N implies that sn > M = 1/ε. Since each sn is positive we have

,

whenever n ≥ N. Thus lim (1/sn) = 0.