On Subsequences and Divergence of Sequences

Question

We know that if a sequence X has two convergent subsequences, whose limits are not equal, then X is divergent. For example: $(-1)^n$

However, is it true that for a given divergent sequence, we can necessarily find at least two convergent subsequences whose limits are unequal? Please help in proving or disproving this claim!

Thanks in advance!

Answer 1

No, take the sequence $a_n=n$. It does not have any convergent subsequences.

EDIT: Let $(a_{n_k})_{k\in\mathbb N}$ be an arbitrary subsequence of $(a_n)_{n\in\mathbb N}$. Then $n_k \geq k$ for all $k\in\mathbb{N}$ since $(n_k)_{k\in\mathbb N}$ is a strictly monotonically increasing sequence. This implies $a_{n_k}\geq k$ for all $k\in\mathbb N$. Therefore the subsequence $(a_{n_k})_{k\in\mathbb N}$ does not converge.