Condition (iii) is also sufficient. If $|a_n|$ did not tend to infinity, some $M>0$ would contain infinitely many terms with $|a_n|\leq M$. Those terms form a <bounded subsequence>, and the <Bolzano-Weierstrass theorem> would give it a convergent subsequence, contrary to the hypothesis. Hence $|a_n|\to\infty$ and
$$
\boxed{\frac1{a_n}\longrightarrow0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 1 / 3E / iii / Solution

A bounded subsequence is a <subsequence> whose terms all lie in one fixed bounded set.

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