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

Condition (iii) is also sufficient. If $|a_n|$ did not tend to infinity, some $M>0$ would contain infinitely many terms with $|a_n|\le M$. 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