Suppose that $(a_{n})_{n=1}^{\infty}$ is a sequence such that $ a_{2n-1}\leq a_{2n+1} \leq a_{2n+2} \leq a_{2n+4}$. Prove that if $(a_{n})$ converges, $\lim_{n \rightarrow \infty} (a_{n}-a_{n+1}) = 0$.
Edit: so my basic idea of a solution involves taking the subsequence of even-indexed terms, which can be shown to be monotonously increasing and the subsequence of odd-indexed terms and saying that if the limit converges to a limit $L$.
Let $(a_n)$ converge. Then: $$ \lim_{n \to \infty}|(a_{n+1}-a_n)|\leq \lim_{n \to \infty} |a_n-a|+\lim_{n \to \infty} |a_{n+1}-a|=0 $$ This is true since $a_n$ and $a_{n+1}$ converge to the same limit. If you assume convergence of $a_n$, you dont need to "monoticity requirement" at all.
Additional note: $$ |a_{n+1}-a_n|\leq |a_{n+1}-a|+|a-a_n| $$ by the triangle inequality.