Question

Give an example of a divergent sequence which has two convergent subsequences.

Justify your claim.

Answer 1

Sequence $a_n = \sin(\pi n/2)$ is divergent, since it oscillates between $-1, 0, 1$ as $n\to\infty$. However, if we take only even $n=2k$, subsequence $a_{2k} = \sin(\pi k)$ will converge to as $k\to \infty$.

We can also take $n = 4k+1$ and the subsequence $a_{4k+1} = \sin(2\pi k+\pi/2)$ will converge to $1$ as $k\to \infty$.

Answer. $a_n = \sin(\pi n/2)$.