I have a rather silly question. I have a Hausdorff topological space $X$, and a countable set of continuous maps of $X$ into $X$, say $\{T_n\}_{n\in \mathbb{N}}$.
Given a net $\{T_{b_i}\}_{i\in I}$, $b_i\in\mathbb{N}$ such that $\lim_{i\in I} T_{b_i}(x)$ exists for each $x\in X$ (the limit converges in the product topology) can I reduce this limit to a subsequence (i.e. a subnet that is a sequence)?
(Especially because $b_i$ must be repeating as $\mathbb{N}$ is countable.)
Thanks!