An unbounded, divergent sequence with a convergent subsequence.

Question

thank you

Answer 1

For example a_{2n+1)=0 for all n and a_{2n}=2n for all n. Then the subsequence a_{2n+1} converges to 0 (since it is identically 0) and the whole sequence is unbounded since a_{2n} tends to infinity for n to infinity.