Let $(a\_n)$ be a bounded real sequence. If $a\_n \leq a\_{n+2}$ and $a\_n \leq a\_{n+3}$ for all $n\in \mathbb{N}$ prove that $(a\_n)$ converges. I think I should start by stating that there are at least two subsequences of $(a\_n)$ that converge to $sup\{a\_n\}$ but I'm not sure which direction to go from here. Thank you for your help.