Here, $a_n \uparrow \infty$, $x_n$ is a sequence, it can take negative or positve value. Does the following formula hold? If so, can you provide a proof? \begin{equation} \limsup\frac{\max\{x_1, x_2,...,x_n\}}{a_n}=\limsup\frac{x_n}{a_n} \end{equation} It is easy to prove that $limsup\frac{x_n}{a_n}\leq limsup \frac{\max\{x_1, x_2,...,x_n\}}{a_n}$, since $x_n\leq max\{x_1, x_2,...,x_n\}$. But the other side is hard for me.
This formula doesn't come from any books but from my cognitive snap,so it may not be set up. However, it seems reasonable since the $limsup$ is the max limitation of each sub-sequence. But I cannot proof it in detail.
Thanks in advance for any tips or help in general.