Let $\{f_{\alpha,k}\}_{\alpha\in \mathcal A,k\in \mathbb N}\subseteq \mathbb R$, were $\mathcal A $ is a directed set. Suppose I know that for each $k$, and $\varepsilon >0$, there is $\bar \alpha $ such that
$$|f_{\alpha,k}|<\varepsilon, $$ for all $\alpha \geq \bar \alpha$.
Can I find a monotone final function $h:\Lambda\to \mathcal A$, with $\Lambda$ a directed set, such that for each $\varepsilon >0$ there is a $\bar \lambda$ such that
$$|f_{h(\lambda),k}|<\varepsilon, $$ for every $k\in \mathbb N$ and $\lambda \geq \bar\lambda$?