Let $a_n$ be a bounded sequence which is divergent then there exist convergent and monotone subsequence $a_{n_k}$ and $a_{m_k}$ such that

$\lim_{n \to \infty} |a_{n_k} - a_{m_k}| \gt 0$

Since $a_n$ is bounded and divergent so it must have atleast two limit points(say $l_1$, $l_2$), hence we can find two subsequences $a_{n_k}$ & $a_{m_k}$ such that $a_{n_k}$ converges to $l_1$ and $a_{m_k}$ to $l_2$.

Now since $a_{n_k} - a_{m_k}$ $\to$ $l_1 - l_2$

$\implies$ $|a_{n_k} - a_{m_k}|$ $\to$ $|l_1- l_2|$

I have two questions at this point how do I prove that given limit is positive and secondly how do I ensure that the sequences are monotonic

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    $\begingroup$ Consider $\limsup_{n\to\infty} a_n$ and $\liminf_{n\to\infty} a_n$, which are finite because $a_n$ is bounded and unequal because $a_n$ is not convergent. $\endgroup$ – Math1000 May 14 at 14:22
  • $\begingroup$ To call $\;l_1,\,l_2\;$ "limits" of the sequence can be misleading. Perhaps a better name would be "partial limits". $\endgroup$ – DonAntonio May 14 at 14:29
  • $\begingroup$ @Math1000 okay so if i take$ a_nk = \limsup a_n$ and $a_mk = \liminf a_n$ then given sequences are monotonic and the corresponding difference $|a_nk - a_mk|$ will be positive is this proof correct ? $\endgroup$ – sat091 May 14 at 14:40

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