I want to prove or disprove the following
Let $(a_n)$ be a bounded sequence such that
$$\lim_{n\to\infty}(a_{n+1}-a_n)=0$$
Then $(a_n)$ is convergent.
My idea to prove that the proposition above is true is as follows: since $(a_n)$ is bounded, it has limit points. Then, by assuming that the sequence is not convergent, there are two different limit points $\alpha$ and $\beta$ and subsequences of $(a_n)$: $(a_{k_n})$ and $(a_{j_n})$ such that $a_{k_n}\to \alpha$ and $a_{j_n}\to\beta$. Now, I tried to build a subsequence $(a_{p_n})$ of $(a_n)$ with the property $p_{2n+1}=p_{2n}+1$ such that $(a_{p_{2n+1}})$ is a subsequence of $(a_{k_n})$ and $(a_{p_{2n}})$ is a subsequence of $(a_{j_n})$ thus there will exist a subsequence of the sequence $(a_{n+1}-a_{n})$ convergent to $\alpha-\beta$, which would be a contradiction. My problem was that I could not prove that you can always build the subsequece $(a_{p_n})$.
Best Answer
No: consider the sequence $$1,0,\frac12,1,\frac23,\frac 13,0,\frac 14,\frac24,\frac 34,1,\frac 45,\frac35,\frac25,\frac15,0,\ldots$$