Suppose a situation whereby I know for some bounded collection $a_{m,n}$ of real numbers the following:
For any $n \geq 1$, $\displaystyle\liminf_{m\to\infty} a_{m,n} \leq B_{n}$ and we can choose the $B_{n}$ such that $\sup_{n\geq 1}B_{n} <\infty$.
Can I conclude that $\displaystyle\liminf_{m \to \infty} \left(\sup_{n\geq 1}a_{m,n}\right) \leq \sup_{n\geq 1} B_{n}$?
Counter-examples also appreciated.