Suppose we have a uniformly bounded non-negative double indexed sequence $\{x_{m,n}\}_{m,n}$ such that $\forall m\in\mathbb N$ $\lim_{n\rightarrow\infty}x_{m,n}=1$. Then what can be said about the convergence of $\max_{m=1}^{n}x_{m,n}$ ? Or what conditions or assumptions need to be made for convergence of the $\max_{m=1}^{n}x_{m,n}$ ?
Also suppose we don't have the assumption about limit as above. Then what conditions on $\{x_{m,n}\}_{m,n}$ is needed for the convergence of $\max_{m=1}^{n}x_{m,n}$ ?
I'm stuck on this issue in my thesis, and cannot find any resource regarding this. Any help is appreciated.