Define $a_{2k}=0$ and $a_{2k-1}=k$. The odd-indexed terms are unbounded, while the even-indexed <subsequence> is identically zero and hence converges to zero. This gives \b[an unbounded <sequence> with a convergent <subsequence>].

Solution

Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 1 / 3F / c / Solution

 Solution (source code)