# Proving existence of a subsequence

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

• 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. – Math1000 May 14 at 14:22
• To call $\;l_1,\,l_2\;$ "limits" of the sequence can be misleading. Perhaps a better name would be "partial limits". – DonAntonio May 14 at 14:29
• @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 ? – sat091 May 14 at 14:40